Splitting theorems for certain equivariant spectra
L. Gaunce Lewis, Jr.
Author address:
Math Dept, Syracuse University, Syracuse NY 132441150
Email address: gaunce@ichthus.syr.edu
1991 Mathematics Subject Classification.
Primary 55M35, 55N91, 53P42, 55P91, 57S15;
Secondary 55N20 55Q10 55Q91 55R12.
Abstract.Let G be a compact Lie group, be a normal subgroup of G,
G = G=, X be a Gspace and Y be a Gspace. There are a number of results in
the literature giving a direct sum decomposition of the group [1 X; 1 Y ]G of
equivariant stable homotopy classes of maps from X to Y . Here, these results
are extended to a decomposition of the group [B; C]G of equivariant stable
homotopy classes of maps from an arbitrary finite GCW spectrum B to any G
spectrum C carrying a geometric splitting (a new type of structure introduced
here). Any naive Gspectrum, and any spectrum derived from such by a change
of universe functor, carries a geometric splitting. Our decomposition of [B; C]G
is a consequence of the fact that, if C is geometrically split and (F0; F) is a*
*ny
reasonable pair of families of subgroups of G, then there is a splitting of the
cofibre sequence
(EF+ ^ C) __//_(EF0+^ C) __//_(E(F0; F) ^ C)
constructed from the universal spaces for the families. Both the decomposition
of the group [B; C]G and the splitting of the cofibre sequence are proven here
not just for complete Guniverses, but for arbitrary Guniverses.
Various technical results about incomplete Guniverses that should be of
independent interest are also included in this paper. These include versions
of the Adams and Wirthm"uller isomorphisms for incomplete universes. Also
included is a vanishing theorem for the fixedpoint spectrum (E(F0; F) ^ C)
which gives computational force to the intuition that what really matters about
a Guniverse U is which orbits G=H embed as Gspaces in U.
Contents
Introduction 1
Notational conventions 5
Part 1. Geometrically Split Spectra 7
Section 1. The notion of a geometrically split Gspectrum 8
Section 2. Geometrically split Gspectra and Gfixedpoint spectra 11
Section 3. Geometrically split Gspectra and fixedpoint spectra 15
Section 4. Geometrically split spectra and finite groups 21
Section 5. The stable orbit category for an incomplete universe 25
Part 2. A Toolkit for Incomplete Universes 29
Section 6. A vanishing theorem for fixedpoint spectra 30
Section 7. SpanierWhitehead duality and incomplete universes 32
Section 8. Change of group functors and families of subgroups 33
Section 9. Change of universe functors and families of subgroups 38
Section 10. The geometric fixedpoint functor for incomplete universes 40
Section 11. The Wirthm"uller isomorphism for incomplete universes 44
Section 12. An introduction to the Adams isomorphism for incomplete
universes 50
Part 3. The Longer Proofs 55
Section 13. The proof of Proposition 3.10 and its consequences 56
Section 14. The proofs of the main splitting theorems 61
Section 15. The proof of the sharp Wirthm"uller isomorphism theorem 65
Section 16. The proof of the Adams isomorphism theorem for incomplete
universes 68
Section 17. The Adams transfer for incomplete universes 75
iii
iv CONTENTS
Acknowledgments 87
Bibliography 88
Introduction
This is a memoir in three parts. The first two parts present two closely rel*
*ated
collections of results on equivariant stable homotopy theory. The last part con*
*tains
the details of some of the messier proofs of the results stated in the first tw*
*o parts.
Roughly speaking, the central theme of the first part is that a family of we*
*ll
known splitting results [46, 1013, 23, 24, 28] about the fixedpoint spectra *
*of
equivariant suspension spectra can be generalized to a significantly larger cla*
*ss of
spectra. This larger class includes all spectra obtained by change of universe *
*func
tors from spectra indexed on a trivial universe. The generalizations of these r*
*esults,
and the class of spectra to which they apply, are best understood by considering
the relation between two types of equivariant fixedpoint spectra. Assume that G
is a compact Lie group, is a (closed) subgroup of G, and C is a Gspectrum
indexed on a Guniverse U. The two different fixedpoint spectra, C and C,
associated to C are both WG spectra indexed on the fixed universe U . The
spectrum C is referred to here simply as the fixedpoint spectrum of C. It co*
*uld
also be described as the categorical fixedpoint spectrum because it is charact*
*erized
by an adjunction, a special case of which identifies the equivariant stable hom*
*otopy
group [G= ^ Sn; C]G with the nonequivariant stable homotopy group [Sn; C ].
This identification is one of the primary reasons for the importance of this ty*
*pe of
fixedpoint spectra.
The other type C of a fixedpoint spectrum is referred to here as a geometr*
*ic
fixedpoint spectrum. These spectra first became widely known in the context of
the proof of the Segal conjecture. Frank Adams, in particular, discussed them
extensively in [1]. They are best understood by considering the case in which C*
* is
the equivariant suspension spectrum 1UY associated to a based Gspace Y . In th*
*is
case, there is an obvious WG spectrum, 1U (Y ), indexed on U which deserves
to be considered as some sort of fixedpoint spectrum of 1UY . However, it is
not immediately obvious that the assignment of 1U (Y ) to 1UY gives a well
defined functor from the full subcategory of the Gstable category generated by*
* the
suspension spectra to the WG stable category. Establishing the functoriality *
*of
this construction in this context only raises the question of whether the const*
*ruction
can be extended to give a functor from the entire Gstable category to the WG 
stable category. In fact, the geometric fixedpoint functor C provides such an
extension. However, here we encounter a surprise. There is a canonical map
i : 1U (Y )___//(1UY );
and it would be natural to expect that this map extends to a natural transforma*
*tion
from C to C . In fact, the only known natural transformation relating these
two functors is a map
! : C __//_ (C)
1
2 INTRODUCTION
going in the opposite direction. The connection between this map and the map i *
* in
the case where C = 1UY is that the composite ! Oi is a weak WG equivalence.
Thus, when C = 1UY , C splits off from C as a wedge summand.
The starting point for the research which led to this memoir was the observa*
*tion
that the proofs of the family of splitting results for suspension spectra menti*
*oned
earlier make use of nothing more than the fact that, for each subgroup , C
splits off from C as a wedge summand. Thus, these splitting results extend to
any Gspectrum C having a right inverse for the map ! for each subgroup of G.
This observation, together with a result indicating that the class of spectra h*
*aving
these right inverses is significantly larger than the class of suspension spect*
*ra, form
the foundation for the work presented here.
It would be natural to call a spectrum having this collection of right inver*
*ses
a split spectrum. However, that term has already been used to describe another
class of spectra [23]. The spectra with this new structure are therefore called
geometrically split spectra, emphasizing the role that the geometric fixedpoint
functors play in their definition. However, since the previously defined split *
*spectra
play no role here, geometrically split spectra are often referred to here simpl*
*y as
split spectra.
In the first part of this memoir, we introduce the precise definition of thi*
*s notion
of a geometrically split Gspectrum, provide a list of examples of such spectra*
*, and
describe a few spectra that are known not to be geometrically split. Also, we s*
*tate
two main splitting theorems for geometrically split spectra. One of these is o*
*ur
generalization of the family of splitting theorems for suspension spectra that *
*have
already appeared in the literature [46, 1013, 23, 24, 28]. For a finite group*
* G,
the simplest case of this extension has roughly the form:
Theorem. Let C be a geometrically split Gspectrum indexed on a complete
Guniverse U. Then there is a weak equivalence
W
CG ' EW + ^W C
()
of nonequivariant spectra indexed on UG . The wedge on the right is indexed over
the Gconjugacy classes () of subgroups of G. This weak equivalence is natural
in C with respect to maps which preserve the geometric splittings.
The earlier forms of this splitting result have been used for a variety of t*
*asks
such as identifying the equivariant zeroth stable stem as the Burnside ring, st*
*udying
Mackey functors, and proving the Segal conjecture. Our generalization can be us*
*ed
to identify the E2terms of some of the suspension and change of universe spect*
*ral
sequences introduced in [20]. It also has applications to the study of Mackey
functors [21].
Our other main splitting theorem seems to be essentially new. It applies to
certain cofibre sequences and long exact sequences in homology derived from the
cofibre sequence
EF+ ___//EF0+___//E(F0; F)
associated to a pair of Gfamilies (F0; F). In its simplest form, our splitting*
* theorem
asserts that, if C is a geometrically split Gspectrum, then the cofibre sequen*
*ce
(EF+ ^ C)G ___//(EF0+^ C)G ___//(E(F0; F) ^ C)G ;
INTRODUCTION 3
derived from the sequence above, is split _ that is, that (EF0+^C)G is the wedg*
*e of
the other two spectra. Given the important role that cofibre sequences associat*
*ed
to pairs of Gfamilies have played in equivariant homotopy theory, our splitting
results for these cofibre sequences ought to have a wide variety of application*
*s.
The impetus behind the second part of this memoir is the large number of
connections between geometrically split spectra and change of universe functors.
In the first place, change of universe functors are used in the construction of*
* all the
known geometrically split spectra. Further, one main application of our splitti*
*ng
results is to the study of change of universe functors. In particular, as we h*
*ave
already noted, these results can be used to compute the E2terms of some of the
change of universe spectral sequences introduced in [20]. Moreover, some of our
results, such as Corollaries 2.5 and 3.4 and Remark 3.9, provide explicit formu*
*lae
for the homotopy groups of the spectra in the image of certain change of univer*
*se
functors. Thus, this memoir provides tools for studying change of universe func*
*tors
which should be thought of as complementary to the tools developed in [1820].
The desire to use our splitting results for the study of change of universe *
*func
tors made it imperative that all of them be proven for as wide a range of incom*
*plete
Guniverses as possible. The effort to prove these splittings in this generalit*
*y led
to the discovery of a number of technical results on incomplete universes. These
technical results are presented in the second part of this memoir. They should *
*be
of interest in their own right. Moreover, most of them are needed for the proof*
*s of
the results stated in the first part.
One of the most noteworthy of our results on incomplete universes gives a
contractibility criterion for spectra of the form (E(F0; F) ^ C) . This result *
*gives
computational force to the suggestion offered by Theorem 1.2(b) of [20] that wh*
*at
really matters about a Guniverse U is which orbits G= embed in U as Gspaces.
In particular, it is the key to showing that, if the orbit G= does not embed in*
* U
as Gspace, then it cannot have a properly behaved SpanierWhitehead dual. Two
other items of special interest in the second part are treatments of the Adams *
*and
Wirthm"uller isomorphisms for incomplete universes.
One common thread in our generalizations of results to incomplete universes
is the role played by new families of subgroups of the ambient group G. A typic*
*al
extension associates a Gfamily F(U) to each incomplete Guniverse U and asserts
that the conclusion applicable to all Gspectra indexed on a complete universe *
*holds
only for F(U)spectra indexed on U. Providing insight into these new families, *
*and
tools for manipulating them, is therefore one of the goals of the second part.
The first part of this memoir is organized in a somewhat eccentric fashion
because it is intended to serve two audiences. Some special cases of the splitt*
*ing
theorems mentioned above have a variety of applications and should be of intere*
*st to
a broad audience. However, the most general versions of these splitting results*
* are
rather technical in form and of interest only to people looking closely at equi*
*variant
stable homotopy theory. Thus, after giving a precise definition of the notion o*
*f a
geometrically split spectrum in section one, we present the special cases of br*
*oadest
interest in section two and the first half of section three. In the rest of the*
* third
section, we state our splitting theorems in their full generality. This approac*
*h is
somewhat inefficient, but it should make the results more accessible. In the fo*
*urth
section, the simplifications of our results which occur when the group G is fin*
*ite are
discussed. The implications of our results for the stable orbit category associ*
*ated to
an incomplete universe are discussed in Section 5. This material is used in [14*
*, 21]
4 INTRODUCTION
and in our discussion of the Adams and Wirthm"uller isomorphisms. The proofs of
some of the results stated in this part are long and technical; these proofs ar*
*e given
in Part 3.
Almost all of the second part of this memoir should be regarded as a supplem*
*ent
to [24] which extends to incomplete universes various results proven in [24] on*
*ly
for complete universes. The first four sections of Part 2 contain a discussion *
*of fam
ilies of subgroups, and pairs of families of subgroups, in the context of incom*
*plete
universes; these are essentially a supplement to Sections II.1, II.2, and V.7 o*
*f [24].
The fifth section is devoted to the behavior of the geometric fixedpoint funct*
*or
, and extends Section II.9 of [24]. The sixth and seventh sections provide in
troductions to the Wirthm"uller and Adams isomorphisms for incomplete universes,
and so supplement Sections II.6 and II.7 of [24]. The proofs of most of the res*
*ults
in this part are rather short and are given immediately following the statement*
*s.
However, a few longer proofs, such as the proof of the Adams isomorphism theorem
for an incomplete universe, are delayed until the third part.
In addition to containing the proofs of various results from the first two p*
*arts,
the third part also contains a construction of the Adams transfer in the context
of an incomplete universe. This should be of interest in its own right because *
*the
procedures for constructing transfers introduced in Chapter 4 of [24] turn out *
*to be
inadequate for constructing many of the transfers that ought to exist in the co*
*ntext
of an incomplete universe. The construction presented here is a small step towa*
*rd
supplying these missing transfers.
Notational conventions
Much of our notation, and many basic facts about equivariant stable homotopy
theory, are drawn from the first two chapters of [24]. In particular, groups a*
*re
assumed to be compact Lie groups, and subgroups are understood to be closed.
Throughout this memoir, we work with an ambient compact Lie group G, a normal
subgroup of G (which is G itself in one special case), and the associated quot*
*ient
group G=. Our notation is complicated by the fact that, in both the statements
of our results and their proofs, we must work with a collection of subgroups of
G which have no particular relation to , a collection of subgroups of , and a
collection of subgroups of G=. This large assortment of subgroups creates both a
shortage of symbols and a certain amount of confusion about which group contains
each subgroup. To resolve these problems, subgroups of G which have no particul*
*ar
relation to are denoted by the letters immediately following G, such as H and
J. Subgroups of are denoted by capital Greek letters such as and . The
quotient group G= is denoted G, and its various subgroups are denoted by other
script capitals. Typically, whenever subgroups H of G and H of G appear in the
same context, H is the inverse image of H under the projection map G __//_G. The
notation K H indicates that K is a subgroup of H. The set of subgroups of
G which are Gconjugate to H is denoted (H)G , and the notation (K)G (H)G
indicates that K is subconjugate (in G) to H.
All topological spaces are assumed to be compactly generated, weak Hausdorff
spaces [15, 27, 30]. All Gspaces are left Gspaces. Whenever possible, the pre*
*fix
G is omitted from our notations, so that by spaces, subspaces, spectra, maps, e*
*tc.,
we mean Gspaces, subGspaces, Gspectra, Gmaps, etc. If H is a subgroup of G
and Y is a Gspace, then Y H is the Hfixed subspace of Y . A based Gspace is a
Gspace Y together with a specified basepoint, which is required to be in Y G. *
*If
X is an unbased Gspace, then X+ denotes the disjoint union of X and a Gtrivial
basepoint. Our Gspectra are indexed on a Guniverse U as defined in section I.2
of [24]. If C and D are Gspectra indexed on U, then the set of maps from C to D
in the Gstable category of spectra indexed on U is denoted [C; D]UG.
All Grepresentations are assumed to have a Ginvariant inner product. If W
is a Grepresentation, then DW , SW , and SW denote its unit disk, its unit sp*
*here,
and its onepoint compactification, respectively. The basepoint of SW is the p*
*oint
at infinity. If V is a subrepresentation of the Grepresentation W , then W  V
denotes the orthogonal complement of V in W . It is therefore an actual, and no*
*t a
virtual, Grepresentation.
If is a subgroup of G, then NG and WG = NG = are the normalizer of
in G and the associated Weyl group of . If U is a Guniverse and is another
subgroup of G such that , then j; : U ___//U is the inclusion of the
fixed universe U into the fixed universe U . If is the trivial subgroup so
5
6 NOTATIONAL CONVENTIONS
that U is just U, then we abbreviate j; to j : U ___//U. Since the inclusi*
*on
j is a linear NG isometry between two NG universes, there are two change of
universe functors
j* : NG SU ___//NG SU and j*: NG SU ___//NG SU
relating the categories NG SU and NG SU of NG spectra indexed on the uni
verses U and U respectively (see section II.1 of [24]). Recall that, if C is*
* a
Gspectrum indexed on U, then the fixedpoint spectrum C of C is a WG 
spectrum indexed on U (see p. 56, [24]). This spectrum is obtained by pulling
C back to the trivial universe U via the functor j* and then passing to fixed
points in an obvious way. When it is necessary to emphasize the change of unive*
*rse
involved in the formation of C , the notation C is expanded to (j*C) . We oft*
*en
regard C as a NG spectrum via the projection NG __//_WG .
Throughout this paper, we work with families, and pairs of families, of sub
groups of some group (usually G or NG ). Whenever two families F0 and F are
referred to as a pair (F0; F), it is assumed that F F0. If EF and EF0 are
the universal spaces associated to these two families, then there is a canonical
map EF __//_EF0. This map, and any map derived from it, such as the map
EF+ ^ Y ___//EF0+^ Y for some space or spectrum Y , is denoted by . The cofibre
of the map : EF+ ___//EF0+is the universal space E(F0; F) of the pair (F0; F).*
* The
canonical inclusion of EF0into the cofibre E(F0; F); and any map derived from t*
*his
inclusion, is denoted . If (F0; F) and (G0; G) are two families of subgroups su*
*ch that
F0 G0and F G, then the canonical maps : EF0 __//_EG0and : EF ___//EG
induce a canonical comparison map E(F0; F) __//_E(G0; G) which is denoted . If
F0is the family of all subgroups of G, then EF0+is just S0 and the universal sp*
*ace
E(F0; F) is usually denoted "EF. In this context, the canonical maps and beco*
*me
the obvious collapse map : EF+ ___//S0and a map : S0___//"EF. For this mem
oir, one of the most important families of subgroups is the NG family FNG []
consisting of those subgroups H of NG which do not contain (see Definition
II.2.3(ii) of [24]). When it is desirable to keep track of in this setting, we*
* denote
the canonical NG map : S0___//"EFNG []by .
If is a subgroup not only of G but also of , as our notation is intended to
suggest, then N is equal to \NG and is therefore a normal subgroup of NG .
It follows that W is a normal subgroup of WG and that there are isomorphisms
WG =W ~=NG =N ~=(NG )=;
where NG is the product of the subgroups and NG . These isomorphisms
allow us to regard WG =W as a subgroup of G = G=. They also allow us to
assign an action of NG to any object which carries a WG =W action. To
emphasize the containment of WG =W in G and to compactify our notation,
the group WG =W is sometimes denoted W.
Part 1
Geometrically Split Spectra
SECTION 1
The notion of a geometrically split Gspectrum
The general notion of a geometrically split Gspectrum C outlined in the int*
*ro
duction actually breaks up into a collection of different possible levels of st*
*ructure
on C. The level of structure required for any particular result about a spectrum
C depends on the indexing universe U for C and the subgroup with respect to
which the fixed points are taken. Here, we present the most general definition *
*of
our notion of a geometrically split spectrum and provide a collection of exampl*
*es
of such spectra.
Definition 1.1.(a) A Gspectrum C indexed on a Guniverse U is (geomet
rically) split at a subgroup of G if there is a WG spectrum C[] indexed on U
and a WG map i : C[] __//_C such that the composite
C[] _i__//C _!__//_ (C)
is a weak WG equivalence of WG spectra indexed on U .
(b) Let D be a collection of subgroups of G. A Gspectrum C indexed on a
Guniverse U is (geometrically) Dsplit if C is geometrically split at for eve*
*ry
subgroup in D. Note that we might as well assume that the collection D is clos*
*ed
under conjugation. We definitely do not assume that D is a family of subgroups
(that is, that it is closed under the passage to subgroups). A variety of diffe*
*rent
collections of subgroups of G are of interest to us. In particular, a Gspectru*
*m C
geometrically split with respect to the family of all subgroups of G is said to*
* be
fully (geometrically) split. For any Guniverse U, a Gspectrum C is Usplit if*
* it is
split with respect to the collection of subgroups of G such that G= embeds in
U as a Gspace. If is a normal subgroup of G, then a a Gspectrum C is split
if it is split with respect to the collection of subgroups of . Yet another col*
*lection
of interest is that of those subgroups of such that = embeds in U as a 
space. A spectrum split with respect to this collection is referred to as (U; )*
*split
spectrum. In the context of studying pairs (F0; F) of families of subgroups of*
* G,
spectra split with respect to the collection of subgroups of contained in F0F*
* are
often of interest. So are spectra split with respect to the collection of subgr*
*oups
of in F0 F such that = embeds in U as a space.
(c) Let D be a collection of subgroups of G. A Gmap f : C __//_C0 between
two Dsplit Gspectra is said to preserve the splittings if, for each subgroup *
* in
D, there is a W map f[] : C[] __//_C0[] making the diagram
f[]
C[] _____//C0[]
i  i0
fflfflf fflffl
C ______//(C0)
8
1. THE NOTION OF A GEOMETRICALLY SPLIT GSPECTRUM 9
commute in the W stable category. Given a Gmap f : C __//_C0, there is only o*
*ne
possible choice for the map f[]. However, for an arbitrary Gmap f, this choice
need not make the diagram commute.
The following propositions and remarks provide a collection of examples of
geometrically split spectra:
Proposition 1.2. Let D be a collection of subgroups of G, and let U be a
Guniverse.
(a) If Y is a Gspace, then the Uindexed suspension spectrum 1UY is fully
split. Moreover, any Gmap h : Y ___//Y 0induces a map 1Uh : 1UY ___//1UY 0
which preserves the splittings. The splitting map i which exhibits 1UY as a
spectrum split at the subgroup is just the canonical map i : 1U (Y )___//(1UY*
* )
introduced in Remarks II.3.14(i) of [24].
(b) If every subgroup of D acts trivially on the indexing universe U, then e*
*very
Gspectrum C indexed on U is Dsplit and every Gmap between Gspectra indexed
on U preserves these splittings. The WG spectrum C[] indexed on U = U which
exhibits the splitting of C at is just C , and the splitting map i : C[] __/*
*/_C
is just the identity map.
(c) The change of universe functor i* associated to a Gisometry i : U ___//U0
takes Dsplit Gspectra to Dsplit Gspectra. Moreover, i* takes maps which pre
serve splittings to maps which preserve splittings. Note in particular that, if*
* U is a
trivial universe, then every spectrum in the image of i* is fully split and eve*
*ry map
in the image of i* preserves splittings. If C[] is the WG spectrum indexed on U
which exhibits the splitting of C at , then i*C[] is the WG spectrum indexed
on (U0) which exhibits the splitting of i*C at . The splitting map for i*C at
is just the composite
i*i
i*C[] _____//_i*(C ) __//_(i*C) ;
in which the second map is the canonical map. In this discussion, i* is just t*
*he
change of universe functor associated to the induced linear isometry i : U __/*
*/_(U0)
between the fixedpoint universes.
(d) If Y is a Gspace and C is a Dsplit Gspectrum, then Y ^ C is Dsplit.
Moreover, if h : Y ___//Y 0is a Gmap between Gspaces and f : C __//_C0 is a G
map between Dsplit Gspectra which preserves the splittings, then the induced *
*map
h^f : Y ^C __//_Y 0^C0preserves the splittings. If C[] is the WG spectrum which
exhibits the splitting of C at , then Y ^ C[] is the WG spectrum exhibiting t*
*he
splitting of Y ^ C at . The splitting map is the composite
Y ^ C[] _1^i_//_Y ^ C ___//_(Y ^ C) ;
in which is the canonical map of Remarks II.3.14(ii) of [24].
(e) If C and D are Dsplit Gspectra, then C ^D is Dsplit. Moreover, any pair
of Gmaps f : C __//_C0 and g : D __//_D0 between Dsplit Gspectra which prese*
*rve
the splittings induces a map f ^ g : C ^ D __//_C0^ D0which preserves the split*
*tings.
If C[] and D[] are the WG spectra which exhibit the splittings of C and D at
, then C[] ^ D[] is the WG spectrum exhibiting the splitting of C ^ D at .
The splitting map is the composite
i;C ^i;D !
C[] ^ D[] _________//_C ^ D ___//_(C ^ D) ;
10 1. THE NOTION OF A GEOMETRICALLY SPLIT GSPECTRUM
in which ! is the map of Remarks II.3.14(iii) of [24].
(f) If C is a Dsplit Gspectrum, and E is a retract of C in the Gstable cat*
*egory,
then E is also Dsplit. If the maps i : E __//_C and r : C __//_E exhibit E as *
*a retract
of C and the map i : C[] __//_C provides the splitting of C at the subgroup ,
then the composite
i (! Oi )1 i r
E _____// C _________//_C[] ____//C ____//E
provides the required splitting of E at .
(g) If T is a collection of integer primes and C is a Dsplit Gspectrum, then
the localization CT of C at T is also Dsplit.
(h) If C is bounded below and is a Dsplit Gspectrum, then its completion C^p
at any integer prime p is also Dsplit.
Remark 1.3. (a)Every spectrum C is split at the trivial subgroup e since
Ce = eC = C.
(b) Part (d) of the proposition indicates that our splitting theorems for spl*
*it
Gspectra apply not only to these spectra, but also to the homology theories on
Gspaces derived from them.
(c) Recall from [18, 19, 22] that an EilenbergMac Lane Gspectrum indexed
on a universe U is a GCW spectrum C indexed on U such that, for every inte
ger n 6= 0 and every subgroup K of G, ssKnC = 0. If C is an EilenbergMac Lane
Gspectrum, then, for any subgroup of G, C is an EilenbergMac Lane WG 
spectrum. In order for C to be split at , the spectrum C, being a retract of C*
* ,
would also have to be an EilenbergMac Lane WG spectrum. If the indexing uni
verse U is nontrivial, then the spectrum C typically has nonvanishing homotopy
groups in positive dimensions (see Theorem II.9.8 and Proposition II.9.13 in [2*
*4]).
Thus, equivariant EilenbergMac Lane spectra indexed on nontrivial universes a*
*re
typically not split at any subgroup other than the trivial subgroup. However,
the general result about geometric splittings stated below indicates certain sp*
*ecial
EilenbergMac Lane spectra are geometrically split at some nontrivial subgroups.
Proposition 1.4. Let C be a Gspectrum, and be a subgroup of G. If, for
every integer n and every subgroup K of NG which does not contain , ssKnC = 0,
then the canonical map ! : C ___// C is a weak WG equivalence, and so C is
split at .
Proof. Since the existence of a splitting for C at depends only on the str*
*uc
ture of C as a NG spectrum, we may assume that is normal in G. Proposition
II.9.2 and Theorem II.9.8 of [24] then imply that the map ! is a weak WG 
equivalence. Theorem II.9.5 of [24] indicates that this result is actually_a_sp*
*ecial
case of Proposition 1.2(c). __
SECTION 2
Geometrically split Gspectra
and Gfixedpoint spectra
In this section, we explore the implications of a geometric splitting of a G
spectrum C for the structure of various Gfixedpoint spectra associated to C. *
*The
primary implications are two splitting theorems. One of these (Theorem 2.1) see*
*ms
new, but the other (Theorem 2.4) generalizes a large number of splitting result*
*s that
have already appeared in the literature [46, 1013, 23, 24, 28]. The statement
of each of these results consists of two parts: one part is a statement about G
fixedpoint spectra, and the other is the represented form of the spectrumlevel
assertion.
It turns out that, since fixedpoint spectra with respect to proper subgroups
of G are not considered here, there is no real advantage to assuming here that
the indexing Guniverse U for our spectra is complete. Thus, in this section, *
*no
restrictions are imposed on the indexing universe U. Here, also, the Weyl group*
* of
a subgroup of G is denoted W , rather than WG .
Theorem 2.1. Let C be a geometrically Usplit Gspectrum indexed on a G
universe U, and (F0; F) be a pair of Gfamilies. Then the cofibre sequence
(EF+ ^ C)G ___//(EF0+^ C)G ___//(E(F0; F) ^ C)G
is split. Thus, if B is a nonequivariant spectrum indexed on UG and B is regard*
*ed
as a Gspectrum with trivial action, then the portion
G U G 0 U G 0 U
j* B; EF+ ^ CG __//_j* B; EF+ ^ CG___//j* B; E(F ; F) ^GC
of the long exact sequence associated to the pair (F0; F) is a split short exac*
*t se
quence. These two splittings are natural in C with respect to maps of spectra w*
*hich
preserve the geometric splittings. The second splitting is also natural in B.
Remark 2.2. In this theorem, one can weaken the requirement that C be
geometrically Usplit to the requirement that C is geometrically split with res*
*pect
to the collection D of subgroups in F0 F for which the orbit G= embeds in U
as a Gspace.
The theorem above provides us with several examples of spectra that are not
geometrically split.
Example 2.3. (a) In [2], Costenoble describes the structure of the equivaria*
*nt
Thom spectrum MOG and the spectrum mOG representing unoriented geometric
Gbordism for the group G = Z=2. His results include the observation that, if F*
*0is
the family of all subgroups of Z=2 and F is the family consisting only of the t*
*rivial
subgroup, then the maps of equivariant stable homotopy groups induced by the
11
12 2. GEOMETRICALLY SPLIT GSPECTRA AND GFIXEDPOINT SPECTRA
maps
: EF0+^ MOG ___//E(F0; F) ^ MOG
and
: EF0+^ mOG ___//E(F0; F) ^ mOG
are monomorphisms, but not isomorphisms. If either MOG or mOG were geomet
rically fully split, then the corresponding map on homotopy groups would have to
be a split epimorphism, which it clearly cannot be. It follows that MOG and mOG
are not geometrically fully split for any compact Lie group G which contains Z=2
as a subgroup.
(b) Even though completion at an integer prime typically preserves geometric
splittings (see Proposition 1.2(h)), completion at ideals in the Burnside ring *
*need
not. For example, let G be Z=p, for some prime p, and let I be the augmentation
ideal of the Burnside ring of G. The completion (S0)^Iof the sphere Gspectrum *
*at I
is not geometrically split even though the sphere Gspectrum itself is geometri*
*cally
split by Proposition 1.2(a). The nonexistence of a splitting for (S0)^Ican be s*
*een
by examining the cofibre seqence
EG+ ^ (S0)^I_//_(S0)^I//_"EG ^ (S0)^I:
By Theorem 2.1, the associated exact sequence
0^ G 0^ G i 0^j
ssG0 EG+ ^ (S )I __//_ss0 (S )I ___//ss0 E"G ^ (S )I:
of zeroth homotopy groups would have to be a split short exact sequence if (S0)*
*^I
were split. The first two groups in this sequence, and the map between them, are
easily computed. Proposition 3.1 of [7] and the observation that (E"G ^ (S0)^I)*
*G is
G ((S0)^I) make it easy to compute the third group. These computations indicate
that the sequence of homotopy groups has the form
(p;fl) ^ ^
Z _____//_Z Zp __//_Zp;
where fl : Z __//_Z^pis the completion map. Note that the composite of the firs*
*t map
in this sequence with the projection ZZ^p__//Z is not surjective. This observat*
*ion
and the fact that there are no nontrivial maps from Z^pto Z imply that the sequ*
*ence
of homotopy groups cannot be a split short exact sequence.
A bit of additional notation is needed for our second splitting theorem. Let
be a subgroup of G. Then the adjoint representation Ad(W ) of the Weyl
group W of is the tangent space of W at the identity element e with the
W action derived from the conjugation action of W on itself. Let C be a U
split Gspectrum, and assume that the orbit G= embeds in U as a Gspace. The
Usplitting of C provides a W spectrum C[] indexed on U . The spectrum
EW + ^ Ad(W) C[] is a free W spectrum indexed on U . Thus, by Theorem
II.2.6 of [24], there is a W spectrum Z() indexed on UG such that j;G*Z() is
weakly W equivalent to EW + ^ Ad(W) C[]. Since W acts trivially on the
universe UG , there is an orbit spectrum Z()=W indexed on UG . This spectrum
is hereafter denoted EW + ^W Ad(W) C[]. Note that the spectrum C[] in this
construction could be replaced by any other W spectrum D indexed on U . The
resulting nonequivariant spectrum EW + ^W Ad(W) D depends functorially on
D.
2. GEOMETRICALLY SPLIT GSPECTRA AND GFIXEDPOINT SPECTRA 13
Theorem 2.4. Let C be a geometrically Usplit Gspectrum indexed on a G
universe U. Then there is a weak equivalence
W Ad(W)
CG ' EW + ^W C[]
()
of nonequivariant spectra indexed on UG . The indexing in the target of this ma*
*p is
over the Gconjugacy classes () of subgroups of G such that G= embeds as a
Gspace in U. Further, let B be a nonequivariant spectrum indexed on UG which is
regarded as a Gspectrum with trivial action. If either B is a finite CW spectr*
*um
or G is a finite group, then there is an induced isomorphism
G U M h Ad(W) i UG
j* B; CG ~= B; EW + ^W C[] ;
()
in which the direct sum has the same indexing as the wedge in the weak equivale*
*nce.
Both the weak equivalence and the isomorphism are natural in C with respect to
maps which preserve splittings; the isomorphism is also natural in B.
In some important special cases, the spectra EW + ^W Ad(W) C[] appear
ing in Theorem 2.4 have particularly simple descriptions. The resulting simpli*
*fi
cations of the formulae from the theorem are recorded below. The first of these
simplifications is, essentially, a special case of Theorem V.11.1 of [24].
Corollary 2.5. Let B be a nonequivariant spectrum indexed on UG regarded
as a Gspectrum with trivial action, and assume that either B is a finite CW sp*
*ec
trum or G is a finite group.
(a) For any Gspace Y , there is a weak equivalence
W 1 Ad(W)
(1UY )G ' UG (EW + ^W Y )
()
of nonequivariant spectra indexed on UG . This map induces an isomorphism
G 1 U M h 1 Ad(W) i UG
j* B; U Y G ~= B; UG (EW + ^W Y ) :
()
The indexing in the targets of these maps is over the Gconjugacy classes () of
subgroups of G such that G= embeds as a Gspace in U. Both maps are natural
in Y . The isomorphism is also natural in B.
(b) Let X be a Gspectrum indexed on the trivial universe UG . Then there is a
weak equivalence
W Ad(W)
(jG*X)G ' EW + ^W X
()
of nonequivariant spectra indexed on UG . This map induces an isomorphism
G G U M h Ad(W) i UG
j* B; j* XG~= B; EW + ^W X :
()
The indexing in the targets of these maps is over the Gconjugacy classes () of
subgroups of G such that G= embeds as a Gspace in U. Both maps are natural
in X. The isomorphism is also natural in B.
14 2. GEOMETRICALLY SPLIT GSPECTRA AND GFIXEDPOINT SPECTRA
Remark 2.6. As noted in Proposition 1.16 of [20], the second part of the
corollary above can be used to compute the E2terms of some of the change of
universe spectral sequences introduced in Theorem 1.14(b) of [20]. If G is a fi*
*nite
group, X is an EilenbergMac Lane Gspectrum indexed on the trivial universe UG*
* ,
UG
and n is an integer, then Ad(W ) = 0 and Sn; EW + ^W X is just the
ordinary group homology Hn(W ; ss0X ) of the group W . Here, the action of
W on the coefficient group ss0X is just the natural one derived from the acti*
*on of
W on X . Thus, when G is finite and X is an EilenbergMac Lane Gspectrum,
the entire righthand side of the isomorphism in Corollary 2.5(b) can be descri*
*bed
in terms of ordinary group homology.
Our second splitting theorem provides some insight into the relation between
change of universe functors and fixedpoint spectra.
Corollary 2.7. Let C be a Gspectrum indexed on a universe U, i : U __//_U0
be a linear Gisometry, and iG : UG ___//(U0)Gbe the induced linear isometry on
the fixedpoint universes. If C is geometrically U0split, then the natural map
ffi : iG*(CG_)_//(i*C)G
is a split monomorphism in the stable category of spectra indexed on (U0)G .
A key ingredient in the proofs of all the splitting results stated above is *
*an
observation about the fixedpoint spectrum (E(F0; F) ^ C)G associated to any G
spectrum C and any adjacent pair (F0; F) of families of subgroups of G. Being
applicable to nonsplit spectra, this result ought to be of independent interes*
*t, and
is therefore recorded here. Some precursor of this result lies at the heart of*
* the
proofs of most of the predecessors of our splitting theorems.
Proposition 2.8. Let C be a Gspectrum indexed on a Guniverse U, (F0; F) be
an adjacent pair of families of subgroups of G, and B be a nonequivariant spect*
*rum
indexed on UG . Also, let be a subgroup of G such that F0 F = (). If G=
does not embed in U as a Gspace, then (E(F0; F) ^ C)G is weakly contractible, *
*and
jG*B; E(F0; F) ^UCG= 0. Otherwise, there is a natural isomorphism
G 0 U h Ad(W) i UG
j* B; E(F ; F) ^GC~= B; EW + ^W C :
This isomorphism arises from a natural weak equivalence
(E(F0; F) ^ C)G ' EW + ^W Ad(W) C
of nonequivariant spectra indexed on UG .
SECTION 3
Geometrically split Gspectra
and fixedpoint spectra
Here, the results on Gfixedpoint spectra presented in the previous section*
* are
generalized to results on fixedpoint spectra, where is a proper subgroup of *
*G.
In this broader context, a variety of rather unpleasant complications arise whe*
*n the
indexing universe U is incomplete. Thus, in this section, some results are stat*
*ed
first for the special case in which the universe U is complete. This is the mo*
*st
interesting special case, and the results available for it are much cleaner and*
* more
satisfying than those available in the general case.
If C is a Gspectrum indexed on a Guniverse U and is a proper subgroup of
G, then the fixedpoint spectrum C is a WG spectrum indexed on U . Any
elements of G not in NG have no influence on C , so we lose nothing by assumi*
*ng
that is normal in G. In this case, WG is just G=, which is hereafter denoted
G.
The results of Theorem 2.1 on the standard cofibre sequences associated to
a pair of families (F0; F) extend to the context of fixedpoint spectra only i*
*f a
restriction is placed on the pair. This restriction is expressed as a closure c*
*ondition
on the difference F0 F. The closure condition seems to be a natural one, and
appears repeatedly in the study of fixedpoint spectra.
Definition 3.1.A collection D of subgroups of G (such as F0 F) is closed
if, whenever H and K are subgroups of G such that H \ = K \ and H 2 D,
then K is also in D. Note that, if D is a closed collection of subgroups of G,*
* then
H 2 D if and only if H \ 2 D. Thus, a closed collection of subgroups D is
completely determined by the set of subgroups of such that is in D. A pair
(F0; F) of families of subgroups of G is said to be a closed pair if the diffe*
*rence
F0 F is closed.
Theorem 3.2. Let be a normal subgroup of the compact Lie group G, G =
G=, C be a Gspectrum indexed on a Guniverse U, and (F0; F) be a closed pair
of Gfamilies. If C is geometrically split at every subgroup of such that is
in F0 F and = embeds in U as a space, then the cofibre sequence
(EF+ ^ C) ___//(EF0+^ C) ___//(E(F0; F) ^ C)
is split in the Gstable category. Thus, if B is a Gspectrum indexed on U , t*
*hen
the portion
U 0 U 0 U
j* B; EF+ ^ CG __//_j* B; EF+ ^ CG___//j* B; E(F ; F) ^GC
of the long exact sequence associated to the pair (F0; F) is a split short exac*
*t se
quence. These two splittings are natural in C with respect to maps of spectra w*
*hich
preserve the geometric splittings. Also, the second splitting is natural in B.
15
16 3. GEOMETRICALLY SPLIT GSPECTRA AND FIXEDPOINT SPECTRA
The generalization of Theorem 2.4 to fixedpoint spectra is a result in whi*
*ch
rather unpleasant complications arise when the universe U is incomplete. Thus, *
*be
fore stating the full generalization, we consider first the complete universe c*
*ase. In
order to state even this generalization, we must introduce the appropriate repl*
*ace
ment for the construction EW + ^W Ad(W) C[]. In the context of studying
fixedpoint spectra, the significant subgroups of G are those which are also
subgroups of . Recall that, if is a subgroup of , then W is a normal
subgroup of WG , and the quotient group WG =W may be regarded as a sub
group of G = G=. To compactify our notation, we denote the group WG =W
by W. Also recall from Section V.10 of [24] that E(W ; WG ) is the univer
sal W free WG space. Observe that the adjoint representation Ad(W ) of
W actually carries a compatible WG action since the conjugation action of
W on itself extends to a conjugation action of WG on W . If C is a 
split Gspectrum indexed on U and C[] is the associated WG spectrum indexed
on U , then the WG spectrum E(W ; WG )+ ^ Ad(W ) C[] is a W free
WG spectrum indexed on U . Thus, by Theorem II.2.6 of [24], there is a WG 
spectrum Z() indexed on U such that j;* Z() is weakly WG equivalent to
E(W ; WG )+ ^ Ad(W ) C[] . Since W acts trivially on the universe U ,
there is an orbit spectrum Z()=W indexed on U . This Wspectrum is de
noted by E(W ; WG )+ ^W Ad(W ) C[] . Note that the spectrum C[] in
this construction could be replaced by any other WG spectrum D indexed on U .
The resulting Wspectrum E(W ; WG )+ ^W Ad(W ) D depends functori
ally on D.
Theorem 3.3. Let be a normal subgroup of the compact Lie group G, G =
G=, and C be a geometrically split Gspectrum indexed on a complete G
universe U. Then there is a weak equivalence
W i Ad(W ) j
C ' G nW E(W ; WG )+ ^W C[]
()G
of Gspectra indexed on U . The indexing in the target of this map is over the
Gconjugacy classes ()G of subgroups of . Further, let B be a Gspectrum
indexed on U . If either B is a finite GCW spectrum or is a finite group, th*
*en
there is an induced isomorphism
U M h Ad(W ) i U
j* B; CG~= B; E(W ; WG )+ ^W C[] ;
()G W
in which the direct sum has the same indexing as the wedge in the weak equivale*
*nce.
Both the weak equivalence and the isomorphism are natural in C with respect to
maps which preserve the splittings. The isomorphism is also natural in B.
The spectra E(W ; WG )+ ^W Ad(W ) C[], like the analogous spectra
EWG + ^WG Ad(WG) C[] of Section 2, have simpler descriptions in certain
special cases. The analog of Corollary 2.5 stated below records the resulting s*
*im
plifications of the formulae from Theorem 3.3. The first part of this corollar*
*y is
a restatement of Theorem V.11.1 of [24], which is the most general precursor of
Theorems 2.4 and 3.3 to be found in the literature.
Corollary 3.4. Let U be a complete Guniverse, and B be a Gspectrum in
dexed on U . Assume that either B is a finite GCW spectrum or is a finite
group.
3. GEOMETRICALLY SPLIT GSPECTRA AND FIXEDPOINT SPECTRA 17
(a) For any Gspace Y , there is a weak equivalence
W 1 i Ad(W ) j
(1UY ) ' U G nW E(W ; WG )+ ^W Y
()
of Gspectra indexed on U . This map induces an isomorphism
1 U M h 1 i Ad(W ) j iU
j* B; U Y G ~= B; U E(W ; WG )+ ^W Y :
() W
The indexing in the targets of these maps is over the Gconjugacy classes ()G of
subgroups of . Both maps are natural in Y . The second is also natural in B.
(b) Let X be a Gspectrum indexed on the trivial Guniverse U . Then there
is a weak equivalence
W i Ad(W ) j
(j* X) ' G nW E(W ; WG )+ ^W X
()
of Gspectra indexed on U . This map induces an isomorphism
U M h Ad(W ) i U
j* B; j* XG~= B; E(W ; WG )+ ^W X :
() W
The indexing in the targets of these maps is over the Gconjugacy classes ()G of
subgroups of . Both maps are natural in X. The second is also natural in B.
Proposition 2.8, which describes certain Gfixedpoint spectra even in the a*
*b
sence of a geometric splitting, is easily extended to fixedpoint spectra in t*
*he
context of a complete Guniverse U. For this extension, we need the analog of t*
*he
notion of an adjacent family appropriate to the context of fixedpoint spectra.
Definition 3.5.Let be a normal subgroup of G, and (F0; F) be a pair of
families of subgroups of G. Then (F0; F) is a adjacent pair if there is a subg*
*roup
of such that
F0 F = {H G  (H \ )G = ()G }:
This adjacent pair is said to be associated to the subgroup . Observe that, if
= G, then a adjacent pair is just an adjacent pair in the usual sense.
Proposition 3.6. Let C be a Gspectrum indexed on a complete Guniverse
U, be a normal subgroup of G, G = G=, and B be a Gspectrum indexed on
U . Also, let be a subgroup of , and (F0; F) be a adjacent pair of families *
*of
subgroups of G associated to . Then there is a natural isomorphism
0 U h Ad(W ) i U
j* B; E(F ; F) ^GC~= B; E(W ; WG )+ ^W C :
W
This isomorphism arises from a natural weak equivalence
i j
(E(F0; F) ^ C) ' G nW E(W ; WG )+ ^W Ad(W ) C
of Gspectra indexed on U .
The difference between Theorem 3.3 and the corresponding result applicable
to incomplete universes arises from the fact that Theorem 3.3 is really a com
posite result consisting of a splitting result for geometrically split spectra *
*coupled
with instances of the Adams and the Wirthm"uller isomorphisms. In an incom
plete universe, neither of these two isomorphisms is commonly available. Thus, *
*the
18 3. GEOMETRICALLY SPLIT GSPECTRA AND FIXEDPOINT SPECTRA
generalization of Theorems 2.4 and 3.3 to a result about the fixedpoint spect*
*ra
of geometrically split spectra indexed on an incomplete universe is a pure spli*
*t
ting results unadorned with either of the two isomorphisms. The conditions under
which the Adams and Wirthm"uller isomorphisms can be applied to improve these
results are discussed in Sections 11 and 12. One unfortunate consequence of the*
* lack
of a Wirthm"uller isomorphism in this context is that the change of group funct*
*or
GnW ? which appears in Theorem 3.3 must be replaced by the less wellunderstood
change of group functor FW [G; ?) introduced in Section II.4 of [24].
In order to state the full generalization of Theorems 2.4 and 3.3, we must
first introduce the appropriate replacement for the universal spaces EW and
E(W ; WG ).
Definition 3.7.Let ae : NG __//_WG be the standard projection. Then
the EWG space E(; ; G; U) is the universal space associated to the family of
subgroups W of WG such that both W \ W = e and, if H = ae1(W ), then
H=H embeds as a Hspace into U. Note that the space E(; ; G; U) is a
W free WG space. It is the empty space unless the orbit = embeds in U as a
space. If U is a complete Guniverse, then E(; ; G; U) is just E(W ; WG ).
If = G and G= embeds in U as a Gspace, then E(; G; G; U) is just the
universal free WG space EWG .
One might expect the space E(; ; G; U) to depend only on the groups W
and WG and the WG universe U . Unfortunately, this need not be the case. The
requirement in the definition that H=H embed as a Hspace into U is too strong,
and may cause the universal space to depend in a very subtle way on the embeddi*
*ng
of in G. The weaker condition that HNG =H embeds as a HNG space into U
is what might have been expected in this definition, but it isn't restrictive e*
*nough
to allow a generalization of Theorems 2.4 and 3.3.
Theorem 3.8. Let be a normal subgroup of the compact Lie group G, G be
G=, C be a Gspectrum indexed on a Guniverse U, and (F0; F) be a closed pair
of Gfamilies. If C is geometrically split at every subgroup of such that is
in F0 F and = embeds in U as a space, then there is a weak equivalence
W * W
(E(F0; F) ^ C) ' FW [G; (j; (E(; ; G; U)+ ^ C[])) )
()G
of Gspectra indexed on U . The indexing in the target of this map is over the*
* G
conjugacy classes ()G of subgroups of such that is in F0F and = embeds
as a space in U. Further, let B be a Gspectrum indexed on U . If either B is*
* a
finite GCW spectrum or is a finite group, then there is an induced isomorphism
0 U M ; U
j* B; E(F ; F) ^GC~= j* B; E(; ; G; U)+ ^ C[]WG ;
()G
in which the direct sum has the same indexing as the wedge in the weak equivale*
*nce.
Both the weak equivalence and the isomorphism are natural in C with respect to
maps which preserve the splittings. The isomorphism is also natural in B.
Remark 3.9. (a)If F0is the family of all subgroups of G and F is the empty
family of subgroups, then E(F0; F) = S0. Thus, applied to this pair of families*
*, the
theorem gives formulae for j* B; CUGand C .
3. GEOMETRICALLY SPLIT GSPECTRA AND FIXEDPOINT SPECTRA 19
(b) If C = 1UY for some Gspace Y , then the terms on the righthand sides of
the formulae in the theorem are given by
FW [G; (j*; (E(; ; G; U)+ ^ C[]))W ) =
FW [G; (j*; (1U E(; ; G; U)+ ^ Y ))W )
and
; U ; 1 U
j* B; E(; ; G; U)+ ^ C[] WG = j* B; U E(; ; G; U)+ ^ Y WG :
(c) If C = j* X for some Gspectrum X indexed on the trivial Guniverse
U , then the spectrum C[] appearing on the righthand sides of the formulae in
the theorem is just j;* (X ).
By comparing Theorem 3.8 with Theorems 2.4 and 3.3, one can come to a fuller
appreciation of the roles played by the Adams and Wirthm"uller isomorphisms in *
*the
two earlier results. Essentially all that is known about the change of groups f*
*unctor
FW [G; ?) comes directly from either its defining adjunction or the Wirthm"ull*
*er iso
morphism. In the context of an incomplete universe, the Wirthm"uller isomorphism
is available only for Wspectra, where W is some poorly understood Wfamily.
Thus, almost nothing is known about the behavior of the functor FW [G; ?) when
the indexing universe is incomplete. One might hope to argue that the spectra to
which this functor is applied in our results happen to be Wspectra. However, t*
*hey
are all fixedpoint spectra, and, even for a wellunderstood family F of subgro*
*ups,
it is hard to determine whether a particular fixedpoint spectrum is an Fspect*
*rum.
If the Adams isomorphism were available, then it would convert these fixedpoint
spectra to orbit spectra. It is somewhat easier to settle the question of whet*
*her
a particular orbit spectrum is an Fspectrum for any given family F. Unfortu
nately, for a typical incomplete universe U, the Adams isomorphism is unlikely *
*to
apply to the spectra appearing on the righthand sides of the formulae in Theor*
*em
3.8. Moreover, even when it does apply, considering the question of whether the
Wirthm"uller isomorphism also applies leads to the conclusion that the Wirthm"u*
*ller
isomorphism is typically unavailable for these spectra. One point of the discus*
*sion
of the Adams and Wirthm"uller isomorphisms for incomplete universes in Sections
11 and 12 is to provide information on when the formulae of Theorem 3.8 can be
simplified using these isomorphisms. Various conditions under which these isomo*
*r
phisms are available are described in Proposition 11.1 and Theorems 11.4, 11.6,
11.8, and 12.2.
One problem caused by the inapplicability of the Adams and Wirthm"uller iso
morphisms in Theorem 3.8 is that there seems to be no reasonable generalization
of Corollary 2.7 to fixedpoint spectra. The difficulty here is that this coro*
*llary
describes the interaction between of a change of universe functor i*, which is a
left adjoint, and the Gfixedpoint functor, which is a right adjoint. The Ada*
*ms
isomorphism plays a critical role in the proof of the corollary because it allo*
*ws us
to replace certain instances of the Gfixedpoint functor with a more easily ha*
*ndled
Gorbit functor. Any generalization of the corollary to fixedpoint spectra wo*
*uld
have to describe the relation of i* to a composite of a fixedpoint functor and*
* a
change of group functor, which is also a right adjoint. In the absence of the A*
*dams
and Wirthm"uller isomorphisms, each of which would relate one of these right ad
joints to a more tractable left adjoint, it is unreasonable to expect to be abl*
*e to
generalize the corollary.
20 3. GEOMETRICALLY SPLIT GSPECTRA AND FIXEDPOINT SPECTRA
Propositions 2.8 and 3.6, which describe certain Gfixedpoint spectra even *
*in
the absence of a geometric splitting, are easily extended to fixedpoint spect*
*ra in
the context of an arbitary Guniverse U. For this extension, recall the notion *
*of a
adjacent pair of Gfamilies from Definition 3.5.
Proposition 3.10. Let C be a Gspectrum indexed on a Guniverse U, be
a normal subgroup of G, G = G=, and B be a Gspectrum indexed on U . Also,
let be a subgroup of , and (F0; F) be a adjacent pair of families of subgroups
of G associated to . Then there is a natural isomorphism
0 U ~= ; U
fl : j* B; E(F ; F) ^GC___//j* B; E(; ; G; U)+ ^ C WG :
This isomorphism arises from a natural weak equivalence
O : (E(F0; F) ^ C)_'_//FW [G; (j*; (E(; ; G; U)+ ^ C))W )
of Gspectra indexed on U .
SECTION 4
Geometrically split spectra and finite groups
If G is a finite group, then the splittings in Theorem 3.3 can be derived di*
*rectly
from Theorem 3.2 and Proposition 3.6 by using a sequence F*G; of families of
subgroups of G. This sequence provides a natural finite filtration for the ent*
*ire
Gstable category. Theorem 3.3 implies that, if C is a split Gspectrum, then
the induced filtration on C is derived from a wedge decomposition of C . In a
similar fashion, Theorem 3.8 can be derived from Theorem 3.2 and Proposition 3.*
*10
using the same sequence of families. Moreover, Theorem 2.4 can be derived from
Theorem 2.1 and Proposition 2.8 using a related, simpler sequence E*Gof families
of subgroups of G. In this section, these two sequence of families are defined,*
* the
filtration quotients of the associated natural filtrations on Gspectra are ana*
*lyzed,
and the relation between these filtrations and geometric splittings is describe*
*d.
The sequence of Gfamilies F*G; is most easily defined in terms of the seque*
*nce
E* for the subgroup of G. The motivation behind the definition of the sequence
of families F*G; is best understood by recalling the notion of a closed collec*
*tion
of subgroups of G from Definition 3.1.
Definition 4.1.(a) Let E0Gbe the set consisting only of the trivial subgroup
{e} of G, and define the families EnG, for n 1, inductively by
EnG= {K G  ifJ is a proper subgroupKof, thenJ 2 En1G}:
Thus, E1Gconsists of the trivial subgroup and all the subgroups of G which are
cyclic of prime order. Since G is finite, there are only finitely many distinct*
* terms
in this sequence. Observe that the filtration E*Gis natural in G in the sense t*
*hat,
if H is a subgroup of G, then, for any n 0, the family EnHof subgroups of H is
just the intersection of EnGwith the set of subgroups of H. Thus, as a Hspace,*
* the
universal EnGspace EEnGis just EEnH.
(b) For each in EnG En1G, let
EnG() = En1G[ ()G :
Then EnG() is a family of subgroups of G such that En1G EnG() EnG. More
over, the pair (EnG(); En1G) is adjacent.
(c) If is a normal subgroup of G, then define the families FnG;, for n 0, by
FnG; = {K G  K \ 2 En}:
The filtration FnG; is natural in G in the sense that, if H G, then FnH;H\ is *
*just
the intersection of FnG; with the set of subgroups of H. Observe that FnG;G= En*
*G.
If C is a Gspectrum indexed on a Guniverse U, then the spectra (EFnG;)+ ^ C,
for n 0, form a finite natural filtration of C with filtration quotients the s*
*pectra
E(FnG;; Fn1G;) ^ C.
21
22 4. GEOMETRICALLY SPLIT SPECTRA AND FINITE GROUPS
(d) For each in En  En1, let
FnG;() = {K G  K \ 2 En()}:
Then FnG;() is a family of subgroups of G such that Fn1G; FnG;() FnG;.
Moreover, the pair (FnG;(); Fn1G;) is adjacent.
We begin our discussion of the relation between our splitting results and the
sequences of families of subgroups introduced above by considering the implicat*
*ions
of these sequences of families for the fixedpoint spectra of spectra indexed *
*on a
complete Guniverse.
Theorem 4.2. Let G be a finite group, G be a normal subgroup, G = G=,
U be a complete Guniverse, C be a Gspectrum indexed on U, and B be a G
spectrum indexed on U . Then
(a) There is an isomorphism
h U iM U
j* B; E(FnG;; Fn1G;) ^~C= B; E(W ; WG )+ ^W C :
G ()G W
Moreover, there is a weak equivalence
W
(E(FnG;; Fn1G;) ^ C) ' G nW E(W ; WG )+ ^W C
()G
of Gspectra indexed on U . The sum and wedge are indexed on the Gconjugacy
classes ()G of subgroups of such that 2 En  En1 . Both maps are natural
in C; the first is also natural in B.
(b) If C is split, then there is an isomorphism
U M h n n1 U i
j* B; CG ~= j* B; E(FG; ; FG; ) ^ C
n G
which is natural in B and in C with respect to maps which preserve the splittin*
*gs.
Moreover, there is a weak equivalence
W n n1
C ' (E(FG; ; FG; ) ^ C)
n
of Gspectra indexed on U which is natural in C with respect to maps which pre
serve the splittings.
Proof. By checking the fixedpoint sets, it is easy to see that, as Gspace*
*s,
W n n1
E(FnG;; Fn1G;) ' E(FG; (); FG; );
()G
where the wedge product is indexed on the set of Gconjugacy classes of subgrou*
*ps
of in En En1. Part (a) follows directly from this decomposition and Proposit*
*ion
3.6. Part (b) is proven by applying Theorem 3.2 inductively to the cofibre sequ*
*ences_
associated to the pairs (FnG;; Fn1G;). _*
*_
Remark 4.3. (a)If C is a split Gspectrum, then the splittings of Theorem
3.3 are the result of combining the splittings provided in the two parts of this
theorem and identifying the spectra C with the spectra C[].
(b) If C is a Gspectrum indexed on U which is not (geometrically) split, then
the description provided in part (a) of the theorem above for the filtration qu*
*otients
of the filtration ((EFnG;)+ ^ C) should be thought of as a partial substitute *
*for
the splittings of Theorem 3.3.
4. GEOMETRICALLY SPLIT SPECTRA AND FINITE GROUPS 23
The analog of Theorem 4.2 for the Gfixedpoint spectrum of a Gspectrum
indexed on any Guniverse U is:
Theorem 4.4. Let G be a finite group, U be a Guniverse, C be a Gspectrum
indexed on U, and B be a nonequivariant spectrum indexed on UG . Regard B as a
Gspectrum with trivial action. Then
(a) There is an isomorphism
G n n1 U M UG
j* B; E(EG ; EG ) ^GC~= B; EWG + ^WG C :
()G
Moreover, there is a weak equivalence
W
(E(EnG; En1G) ^ C)G ' EWG + ^WG C
()G
of nonequivariant spectra indexed on UG . The sum and wedge are indexed on the
Gconjugacy classes () of subgroups of G such that 2 EnG En1Gand G=
embeds as a Gspace in U. Both maps are natural in C; the first is also natural*
* in
B.
(b) If C is Usplit, then there is an isomorphism
G U M G n n1 U
j* B; CG ~= j* B; E(EG ; EG ) ^GC
n
which is natural in B and in C with respect to maps which preserve the splittin*
*gs.
Moreover, there is a weak equivalence
W n n1 G
CG ' (E(EG ; EG ) ^ C)
n
of nonequivariant spectra indexed on UG which is natural in C with respect to m*
*aps
which preserve the splittings.
In our general setting of looking at the fixedpoint spectra of spectra ind*
*exed
on an incomplete universe, we lose the Adams and Wirthm"uller isomorphisms that
are implicitly contained in the first parts of Theorems 4.2 and 4.4. Thus, our *
*most
general result about the sequence of families F*G; is:
Theorem 4.5. Let G be a finite group, G be a normal subgroup, G = G=,
U be a Guniverse, C be a Gspectrum indexed on U, and B be a Gspectrum indexed
on U . Then
(a) There is an isomorphism
h U iM U
j* B; E(FnG;; Fn1G;) ^~C= j;* B; E(; ; G; U)+ ^ C :
G ()G WG
Moreover, there is a weak Gequivalence
W * W
(E(FnG;; Fn1G;) ^ C) ' FW [G; (j; (E(; ; G; U)+ ^ C)) )
()G
of Gspectra indexed on U . The sum and wedge are indexed on the Gconjugacy
classes ()G of subgroups of such that 2 En  En1 and = embeds as a
space in U. Both maps are natural in C; the first is also natural in B.
24 4. GEOMETRICALLY SPLIT SPECTRA AND FINITE GROUPS
(b) If C is (U; )split, then there is an isomorphism
U M h n n1 U i
j* B; CG ~= j* B; E(FG; ; FG; ) ^ C
n G
which is natural in B and in C with respect to maps which preserve the splittin*
*gs.
Moreover, there is a weak Gequivalence
W n n1
C ' (E(FG; ; FG; ) ^ C)
n
which is natural in C with respect to maps which preserve the splittings.
Remark 4.6. (a)Theorems 4.4 and 4.5 follow from Theorems 2.1 and 3.2 and
Propositions 2.8 and 3.10 just as Theorem 4.2 followed from Theorem 3.2 and
Proposition 3.6. The one additional result needed for them is Theorem 6.1, which
allows us to eliminate the summands in the formulae of part (a) of each theorem
corresponding to the orbits which do not embed in U.
(b) If G is a finite group, then Theorems 2.4 and 3.8 follow from Theorems 4.4
and 4.5 just as Theorem 3.3 followed from Theorem 4.2. Moreover, if C is not
geometrically split, then the descriptions provided in part (a) of Theorems 4.4*
* and
4.5 for the filtration quotients of the filtrations ((EEnG)+ ^C)G and ((EFnG;)+*
* ^C)
can be used as a partial substitutes for the splittings of Theorems 2.4 and 3.8.
SECTION 5
The stable orbit category
for an incomplete universe
The precursors of Theorem 2.4 have been used to provide a useful description*
* of
the set of maps in the complete Gstable category between the suspension spectr*
*a of
two orbits G=H and G=K (see Section V.9 of [24]). An extension of this descript*
*ion
to the context of the Gstable category indexed on an incomplete universe has
already been used in [1821]. This extension is also used in our discussion of *
*the
Wirthm"uller and Adams isomorphisms in Sections 11 and 12. Further, it plays a
key role in [14], in which some homological misbehavior of the category of Mack*
*ey
functors for a compact Lie group is described. There are several ways to prove *
*this
extension, but the easiest is to derive it from Theorem 2.4 in the same way tha*
*t the
result for a complete Guniverse was derived from the precursors of this Theore*
*m.
This section is devoted to the statement and proof of this extension.
Although our primary concern is the set of stable maps between orbits, we be*
*gin
with a generalization of Corollary V.9.3 of [24]. This generalization character*
*izes
stable morphism sets in a somewhat broader context. For this characterization,
recall that a spacelevel Gmap f : G=J ___//G=Hbetween two orbits is determined
by the image gH of the identity coset eJ of G=J and that the possible image cos*
*ets
gH are those associated to the elements g 2 G such that g1Jg H.
Theorem 5.1. Let G be a compact Lie group, U be a Guniverse, H be a sub
group of G, and Y be a based Gspace. Then the morphism set [1UG=H+ ; 1UY ]UG
is a free abelian group generated by the equivalence classes [f; k] of certain *
*diagrams
(f; k) of the form
G=H oof_G=J _k_//_Y;
in which f : G=J ___//G=Hand k : G=J ___//Y are spacelevel Gmaps. If f(eJ) =
gH, then the diagram (f; k) represents a generator if the orbit H=(g1Jg) embeds
as an Hspace in U and the map k is not nullGhomotopic. The diagram
0 k0
G=H oof_ G=J0____//Y
is equivalent to the diagram (f; k) if there is a Ghomeomorphism ff : G=J ___/*
*/G=J0
making the spacelevel diagram
fmmG=J OO
vvmmmm  OkO''OO
G=H hhQQ ff o7Y7o
Q0QQQ fflfflk0oooo
f G=J0
commute up to Ghomotopy.
25
26 5. THE STABLE ORBIT CATEGORY FOR AN INCOMPLETE UNIVERSE
Proof. The change of group isomorphism
1 0 1 U
[1UG=H+ ; 1UY ]UG~= U S ; U Y H
reduces the proof to the case in which H = G. This special case follows from Th*
*eo
rem 2.4 in exactly the same way that Corollary V.9.3 of [24] follows from Theor*
*em
V.9.1 of [24]. The embedding condition in the hypotheses of this corollary whic*
*h is
not in Corollary V.9.3 of [24] arises from the difference between the indexing_*
*sets_
for the direct sums which appear in Theorem 2.4 and Theorem V.9.1 of [24]. _*
*_
Remark 5.2. (a)If f(eJ) = gH, then the diagram
G=H oof_G=J _k_//_Y;
represents the composite
cg 1 1Uk 1
1UG=H+ _t_//1UG=(g1Jg)+ ____//U G=J+ ______//U Y;
in which t is the transfer associated to the equivariant bundle G=(g1Jg) __//_*
*G=H
and cg is the usual Ghomeomorphism between orbits associated to conjugate sub
groups.
(b) In each equivalence class of diagrams, the representative (f; k) can be c*
*hosen
so that J H and the map f : G=J ___//G=His the standard projection; that is,
so that f(eJ) = eH.
(c) The equivalence relation on diagrams described in the theorem can also be
obtained by requiring that the left triangle commute exactly and the right tria*
*ngle
Ghomotopy commute.
If the target in the morphism set of the theorem is the suspension spectrum
of an unbased Gspace, then the morphism set can be described in a particularly
simple way.
Corollary 5.3. Let G be a compact Lie group, U be a Guniverse, H be a
subgroup of G, and Z be an unbased Gspace.
(a) The morphism set [1UG=H+ ; 1UZ+ ]UGis a free abelian group whose gener
ators are the equivalence classes of certain diagrams (f; k) of the form
G=H oof_G=J _k_//_Z;
in which f : G=J ___//G=Hand k : G=J ___//Z are spacelevel Gmaps. If f(eJ) =
gH, then the diagram (f; k) represents a generator if the orbit H=(g1Jg) embeds
as an Hspace in U. The diagram
0 k0
G=H oof_ G=J0____//Z
is equivalent to the diagram (f; k) if there is a Ghomeomorphism ff : G=J ___/*
*/G=J0
making the spacelevel diagram
fmmG=J OO
vvmmmm  OkO''OO
G=H hhQQ ff o7Z7o
Q0QQQ fflfflk0oooo
f G=J0
commute up to Ghomotopy.
5. THE STABLE ORBIT CATEGORY FOR AN INCOMPLETE UNIVERSE 27
(b) In particular, for K G, the morphism set [1UG=H+ ; 1UG=K+ ] UGis a
free abelian group whose generators are the equivalence classes of certain diag*
*rams
(f; k) of the form
G=H oof_G=J _k_//_G=K;
in which f : G=J __//_G=Hand k : G=J __//_G=Kare spacelevel Gmaps. If
f(eJ) = gH, then the diagram (f; k) represents a generator if the orbit H=(g1J*
*g)
embeds as an Hspace in U.
Remark 5.4. (a)In both parts of the corollary, each equivalence class of di*
*a
grams has a representative (f; k) for which J H and f(eJ) = eH. In the second
part of the corollary, there is also a representative for which J K and k(eJ) *
*= eK.
However, there may be equivalence classes for which one cannot arrange that J is
in both H and K.
(b) In both parts of the corollary, the desired equivalence relation on diagr*
*ams
may also be obtained by requiring that the left triangle commute exactly and the
right triangle Ghomotopy commute. In the second part, the desired relation may*
* be
obtained by requiring that the right triangle commute exactly and the left tria*
*ngle
commute up to Ghomotopy.
(c) The composite of two stable maps represented by the diagrams
G=H oof_G=J _k_//_G=K
and
G=K oom_G=J0__n_//G=Q
is obtained by considering the spacelevel pullback diagram
0
P __k___//G=J0
m0  m
fflfflk fflffl
G=J _____//G=K;
and applying the standard result about the interaction between transfers and sp*
*ace
level maps in a pullback diagram (see, for example, axiom (A1) in Definition 1.*
*1 of
[17]). Unfortunately, if G is not finite, then the pullback P need not be a dis*
*joint
union of orbits. Thus, a decomposition formula like Theorem IV.6.1 of [24] (or a
double coset formula like Theorem IV.6.3 of [24]) is needed to express the comp*
*os
ite of the maps represented by (f; k) and (m; n) as a sum of the generators of *
*the
morphism set [1UG=H+ ; 1UG=Q+ ] UG. It seems to be quite difficult to determine
the multiplicities with which the various generators appear in this composite. *
*How
ever, it is clear that the only generators which can appear with a nonzero coef*
*ficient
are those represented by diagrams
G=H oop_G=L _q_//G=Q
in which L is an isotropy subgroup of the Gspace P . In particular, L must be
subconjugate to both J and J0. Moreover, if the representatives (f; k) and (m; *
*n)
are chosen so that J; J0 K, k(eJ) = eK, and m(eJ0) = eK, then the allowed
subgroups L are those which appear in the usual double coset decomposition of
J\G=J0.
28 5. THE STABLE ORBIT CATEGORY FOR AN INCOMPLETE UNIVERSE
Part 2
A Toolkit for Incomplete Universes
SECTION 6
A vanishing theorem for fixedpoint spectra
Let U be an incomplete Guniverse, be a normal subgroup of G, and G =
G=. Then there are certain pairs (F0; F) of families of subgroups of G such tha*
*t,
for any Gspectrum D indexed on U, the fixedpoint spectrum (E(F0; F) ^ D)
is weakly Gcontractible. This vanishing result gives computational force to t*
*he
suggestion offered by Theorem 1.2(b) of [20] that what really matters about a G
universe U is which orbits G=H embed in U as Gspaces. It is, moreover, one of
the key tools in the proofs of our splitting theorems.
This section contains the statement and proof of this vanishing result. It a*
*lso
contains a simple lemma about pairs of families of subgroups which allows us to
use the vanishing result to show that certain pairs of families may be replaced*
* by
somewhat smaller pairs of families. We begin with the statement of our vanishing
theorem.
Theorem 6.1. Let be a normal subgroup of G, G = G=, U be a Guniverse,
and (F0; F) be a pair of families of subgroups of G such that, for every subgro*
*up H
in F0 F, the orbit H=H does not embed in U as a Hspace. Then, for any
Gspectrum D indexed on U, the spectrum (E(F0; F)^D) is weakly Gcontractible.
An argument like that used in the proof of Proposition V.7.5 of [24] reduces
the proof of this proposition to the following lemma about adjacent families.
Lemma 6.2. Let H be a subgroup of G, be a normal subgroup of G, G = G=,
and (F0; F) be an adjacent pair of families of subgroups of G such that F0F = *
*(H)G .
If U is a Guniverse such that H=H does not embed in U as a Hspace and D
is a Gspectrum indexed on U, then the spectrum (E(F0; F) ^ D) is weakly G
contractible.
Proof. Let ae : G __//_G be the projection, K be a subgroup of G, K = ae1(*
*K),
U
and n be an integer. The homotopy group G=K+ ^ Sn; (E(F0; F) ^ D)G must
be shown to vanish. Note that G=K and G=K are homeomorphic Gspaces. This
observation, together with the change of group and change of universe adjunctio*
*ns
in chapter II of [24], yields the sequence of isomorphisms:
n 0 U n * 0 U
G=K+ ^ S ; (E(F ; F) ^ D)G ~= [G=K+ ^ S ; j (E(F ; F) ^ D)]G
~= [G=K+ ^ Sn; E(F0; F) ^UD]G
~= [Sn; E(F0; F) ^UD]K:
Here, j* is the change of universe functor derived from the inclusion j : U _*
*__//U.
If (H)G (K)G , then E(F0; F) is Kcontractible and the group [Sn; E(F0; F) ^UD*
*]K
vanishes.
30
6. A VANISHING THEOREM FOR FIXEDPOINT SPECTRA 31
If (H)G (K)G , then we may assume that H K. Clearly, K, and
therefore H is a subgroup of K. Since H=H does not embed in U as a H
space, K=H cannot embed in U as a Kspace. Recall from section I.8 of [24] that
we can approximate D up to a weak Gequivalence by a cofibrant Gprespectrum
"Dindexed on an indexing sequence {Vm }m0 for the universe U. Then
h i
[Sn; E(F0; F) ^UD]K~=colimmSn+Vm ; E(F0; F) ^ "DVm:
K
If n < 0, then, for sufficiently large m, the representation Vm contains a tri*
*vial
representation of dimension at least nso that n + Vm is a well defined repres*
*en
tation. Thus, the colimit ishwell defined even when niis negative. By Lemma V.7*
*.6
of [24], the homotopy group Sn+Vm ; E(F0; F) ^ "DVmalways vanishes because
K __
K=H does not embed as a Kspace in Vm U. __
In the proofs of our main splitting theorems, Theorem 6.1 is frequently used*
* to
show that a pair of families of subgroups can be replaced by somewhat smaller p*
*air.
The following lemma, whose proof requires nothing more than an examination of
fixedpoint sets, facilitates this use of our vanishing theorem.
Lemma 6.3. Let (F2; F1) and (F4; F3) be pairs of families of subgroups of G
such that F2 F4 and F1 = F2 \ F3. Then the pair of canonical Gmaps
E(F2; F1) __//_E(F4; F3) __//_E(F4; F2 [ F3)
between the universal spaces associated to these pairs of families is an equiva*
*riant
cofibre sequence.
SECTION 7
SpanierWhitehead duality
and incomplete universes
It is well known that, if the Gorbit G=H embeds in the Guniverse U, then
G=H has a SpanierWhitehead dual in the equivariant stable category of Gspectra
indexed on U (see, for example, Chapter 3 of [24]) . It has always been assumed*
* that
this sufficient condition was also necessary. Our vanishing theorem for fixedp*
*oint
spectra (Theorem 6.1) provides a way of verifying this assumption.
Proposition 7.1. Let G be a compact Lie group, H be a closed subgroup of
G, and U be a Guniverse into which G=H does not embed as a Gspace. Then
1UG=H+ does not have a SpanierWhitehead dual in the Gstable category of spec
tra indexed on U.
Proof. Let (F0; F) be an adjacent pair of Gfamilies such that F0 F = (H)G*
* .
If there were a Gspectrum Z which was the SpanierWhitehead dual of 1UG=H+ ,
then there would be an isomorphism
[1UG=H+ ; 1UE(F0; F)]UG~=[S0; 1UE(F0; F) ^ Z]UG:
Since G=H doesn't embed as a Gspace in U and F0 F = (H)G , the spectrum
(1UE(F0; F)^Z)G is weakly contractible by Theorem 6.1. This reduces the nonex
istence proof to showing that the lefthand side of the putative equation above*
* is
nonzero. Various change of group isomorphisms can be applied to that side to
obtain
H
[1UG=H+ ; 1UE(F0; F)]UG~=[S0; 1UE(F0; F)]UH~=[S0; (1UE(F0; F))H ]U :
Let F[H] be the Hfamily consisting of all of the proper subgroups of H. Then t*
*he
natural projection
E(F0; F) __//_E(F0; F) ^ "EF[H]
is an Hequivalence, as is shown by a computation of the fixedpoint sets. Coro*
*llary
II.9.9 of [24] therefore indicates that the nonequivariant spectrum (1UE(F0; F)*
*)H is
equivalent to 1UH(E(F0; F)H ). But E(F0; F)H ' S0. Thus, the morphism set on the
lefthand side of the asserted equation can be identified with [S0; S0]UH ~=Z. *
* ___
32
SECTION 8
Change of group functors
and families of subgroups
Here, we study certain pairs (F0; F) of families of subgroups of G, such as
adjacent pairs, which are naturally tied to a single subgroup of G. Our ob
jective is to give conditions under which the universal space E(F0; F) associat*
*ed to
such a pair (F0; F) can be approximated by a space of the form G nNG E(E0; E),
where NG is the normalizer of in G and (E0; E) is a pair of families of sub
groups of NG . Our main result is that, for any Guniverse U, there is a min
imal pair (E0NG <; U>; ENG <; U>) of families of subgroups of NG such that,
for any Gspectrum D indexed on U and any adjacent pair (F0; F) of families
of subgroups of G associated to , the fixedpoint spectrum (E(F0; F) ^ D)
can be approximated, up to a weak Gequivalence, by the fixedpoint spectrum
((G nNG E(E0NG <; U>; ENG <; U>)) ^ D) . This approximation result gener
alizes some results contained in [3] and chapter 7 of [5]. It forms the heart o*
*f the
proofs of Propositions 2.8, 3.6, and 3.10, and also plays a critical role in th*
*e proofs
of our main splitting theorems.
Our approximation result describes the behavior of a particular member of
a family of canonical comparison maps. We begin by introducing this family of
comparison maps and certain key families of subgroups.
For any Gfamily F, let F NG be the NG family {H NG  H 2 F}.
The universal Gspace EF associated with F, when considered as an NG space, is
NG equivalent to E(F NG ). Similarly, the universal Gspace E(F0; F) associa*
*ted
to a pair of Gfamilies (F0; F) is NG equivalent to E(F0 NG ; F NG ). Thu*
*s,
if E is a family of subgroups of NG and F is a family of subgroups of G such
that E F NG , then the canonical NG map : EE+ ___//E(F NG )+induces a
Gmap
" : G nNG EE+ __//_EF+:
Similarly, if (E0; E) is a pair of families of subgroups of NG and (F0; F) is *
*a pair
of families of subgroups of G such that E0 F0 NG and E F NG , then the
canonical NG map : E(E0; E)__//E(F0NG ; F NGin)duces a canonical Gmap
" : G nNG E(E0; E)__//_E(F0;:F)
If E is a family of subgroups of NG , then, by closing the collection E under
conjugation_by elements of G, we obtain_a_family of subgroups of G which is den*
*oted
E. Note that, for any NG family E, E E NG . Thus, there is a canonical Gmap
" : G nNG EE+ __//_E__E+:
33
34 8. CHANGE OF GROUP FUNCTORS AND FAMILIES OF SUBGROUPS
Also, for any pair (E0; E) of families of subgroups of NG , there is a canonical
Gmap
__0 __
" : G nNG E(E0; E)__//_E(E ;:E)
There are two pairs of Gfamilies of subgroups and two pairs of NG families
of subgroups which play a special role in the study of the adjacent pairs (F0;*
* F)
of subgroups of G associated to .
Definition 8.1.(a) Let (F0G<>; FG <>) be the pair of Gfamilies of subgroups
given by
F0G<> = {H G  9g 2 G; gHg1 NG and gHg1 \ }
and
FG <> = {H G  9g 2 G; gHg1 NG and gHg1 \ < }:
If K is a subgroup of G such that K \ = , then K NG , and so K 2
F0G<>. It follows that F0G<>  FG <> = {H G  H \ 2 ()G } so that the
pair (F0G<>; FG <>) is adjacent. This pair is minimal among adjacent pairs
associated to the subgroup of in the sense that, if (F0; F) is another such p*
*air,
then F0G<> F0and FG <> F. There is therefore a comparison Gmap
: E(F0G<>; FG <>)__//_E(F0;;F)
which is unique up to Ghomotopy. Also, the Gfamily F0G<> is minimal among
the Gfamilies containing the set {H G  H \ 2 ()G } in the sense that, if
F0 is any other Gfamily containing this set, then F0G<> F0. Thus, there is a
comparison Gmap : EF0G<> ___//EF0which is unique up to Ghomotopy.
(b) Let (E0NG <>; ENG <>) be the pair of NG families given by
E0NG <> = {H NG  H \ }
and
ENG <> = {H NG  H \ < }:
________ ________
The pair (E0NG <> ; ENG <>) is just the minimal adjacent pair (F0G<>; FG <>)
introduced above. Thus, there is a canonical Gmap
": G nNG E(E0NG <>; ENG <>) __//_E(F0G<>; FG <>):
Moreover, for any adjacent pair (F0; F) of subgroups of G associated to , there
is a canonical comparison map
" : G nNG E(E0NG <>; ENG <>) __//_E(F0;:F)
(c) If U is an incomplete Guniverse, then the NG pair (E0NG <>; ENG <>) is
not as wellbehaved as one might like. A betterbehaved replacement for this pa*
*ir
is given by
E0NG <; U> = {H NG  H \ and H=H Hembeds inU}
and
ENG <; U> = {H NG  H \ < and H=H Hembeds inU}:
__________ __________
Define the Gpair (F0G<; U>; FG <; U>) to be (E0NG <; U>; ENG <; U>). Note that
there are canonical comparison maps
: E(E0NG <; U>; ENG <; U>)__//E(E0NG <>; ENG <>);
8. CHANGE OF GROUP FUNCTORS AND FAMILIES OF SUBGROUPS 35
": G nNG E(E0NG <; U>; ENG <; U>)__//E(F0G<; U>; FG <; U>);
and
: E(F0G<; U>; FG <; U>)_//_E(F0G<>; FG <>):
Moreover, for any adjacent pair (F0; F) of subgroups of G associated to , there
is a canonical comparison map
" : G nNG E(E0NG <; U>; ENG <; U>)__//E(F0; F):
The embedding condition on H=H in Definition 8.1(c), like the embedding
condition in Definition 3.7, is stronger than one might expect. Its significanc*
*e can
be understood by examining Theorem 11.8, which plays an essential role in the
proofs of our main results.
The following lemma, which follows from an examination of fixedpoint sets,
describes an important connection between the universal spaces E(; ; G; U) of
Definition 3.7 and E(E0NG <; U>; ENG <; U>) of Definition 8.1(c).
Lemma 8.2. Let be a subgroup of the normal subgroup of G, and let U be
a Guniverse. If the WG space E(; ; G; U) is regarded as a NG space via the
projection NG __//_WG , then the composite
E(; ; G; U) ___//_EE0NG <; U> ___//_E(E0NG <; U>; ENG <; U>)
is, up to an NG equivalence, the inclusion into E(E0NG <; U>; ENG <; U>) of i*
*ts
fixedpoint set.
Our main approximation result describes the behavior of the final comparison
maps introduced in parts (b) and (c) of Definition 8.1.
Proposition 8.3. Let be a subgroup of the normal subgroup of G, and let
(F0; F) be a adjacent pair of families of subgroups of G associated to . Then *
*the
canonical map
": G nNG E(E0NG <>; ENG <>) __//_E(F0; F)
is a weak Gequivalence. Moreover, if D is a Gspectrum indexed on a Guniverse
U, then the canonical map
(" ^ 1) : ((G nNG E(E0NG <; U>; ENG <; U>)) ^ D)__//(E(F0; F) ^ D)
is a weak Gequivalence.
The first part of the proposition above follows from a general result giving*
* condi
tions under which the canonical comparison map ": G nNG E(E0; E)__//_E(F0; F)
is a Gequivalence.
Proposition 8.4. Let be a subgroup of the normal subgroup of G, and let
(E0; E) be a pair of NG families of subgroups such that
E0 E {H NG  H \ = }:
__0 __
Let E and E be the Gfamilies of subgroups obtained by closing E0 and E under
Gconjugation. If
__
E\ {H G  (H \ )G = ()G } = ;;
then the canonical Gmap
__0 __
": G nNG E(E0; E)__//_E(E ; E)
36 8. CHANGE OF GROUP FUNCTORS AND FAMILIES OF SUBGROUPS
is a Gequivalence.
Proof. Consider the commuting diagram
G nNG E(E0; E)_______"__________//_E(__E0; __E)
1n  
fflffl __ _fflffl__ __
G nNG (E(E0; E) ^ E(E0; E)) E(E 0; E) ^ E(E 0; E)
OO
1n(^1)  1^"
__ fflffl_ ~= __ __ 
G nNG (E(E 0; E) ^ E(E0; E))__//E(E 0; E) ^ G nNG E(E0; E)
in which the bottom horizontal map is the inverse of Ghomeomorphism i of Lemma
II.4.9 of [24]. An examination of its behavior on fixedpoint sets indicates th*
*at the
diagonal map
: E(E0; E)__//E(E0; E) ^ E(E0; E)
appearing in the left column is an NG equivalence. A similar argument indicates
that the other diagonal map appearing in the diagram is a Gequivalence. Thus, *
*to
show that the map " is a Gequivalence, it suffices to show that the map ^ 1 a*
*p
pearing in the left column is an NG equivalence, and that the map 1^ "appearing
in the right column is a Gequivalence.__
The conditions_imposed on E0 E and E in the hypotheses of this proposition
imply that E = E\ E0. It follows_that,_for each H 2 E0 E, the map on fixedpoi*
*nt
sets H : E(E0; E)H__//_E(E 0; E)His a nonequivariant equivalence between spaces
homotopy equivalent to S0. Thus, by Lemma V.7.4 of [24], the map ^ 1 is an
NG equivalence.
To show that 1 ^ " is a_Gequivalence,_it suffices to show that the map "H :
(G nNG E(E0;_E))H__//_E(E_0; E)His_a nonequivariant equivalence for each sub
group H in E0  E. Every H in E0 is Gconjugate to a_subgroup in E0,_so_we
may as well assume that H is in E0. Then, since E = E \ E0 and H 2 E0 E, H
must be in E0 E. Recall from Section II.4 of [24] that there is a natural incl*
*u
sion j : E(E0; E)__//G nNG E(E0; E)such that " O j = . Since we have already
shown that H is a nonequivariant equivalence, to show that "H is a nonequivaria*
*nt
equivalence, it suffices to show that jH is a nonequivariant equivalence.
In fact, we show that, for an appropriately chosen model of E(E0; E), jH is
a homeomorphism. Let g 2 G and x 2 E(E0; E) be such that the equivalence
class [g; x] of the pair (g; x) in G nNG E(E0; E) is in (G nNG E(E0; E))H and
is not the basepoint. Then, for each h 2 H, there is an n 2 NG such that
(hg; x) = (gn; n1x). It follows that n must be in the isotropy subgroup NG x of
x. There is a model for E(E0; E) in which the isotropy groups of all of the poi*
*nts
except the basepoint are in E0. By using this model, we can ensure that NG x is
in E0. Since hg = gn, g1Hg must be a subgroup of NG x and so must be in E0.
By our hypothesis on_E0 E, H \ = , and so g1Hg \ = g1g. But then
g1Hg cannot be in E. Thus, g1Hg 2 E0 E, and g1Hg \ = . It follows that
g1g = so that g 2 NG . By the definition of GnNG E(E0; E), the equivalence
classes [g; x] and [e; g1x] are equal. The map jH is therefore onto. Since_it *
*is also
an inclusion, it is a homeomorphism. __
8. CHANGE OF GROUP FUNCTORS AND FAMILIES OF SUBGROUPS 37
Remark 8.5. Let (E0; E) be a pair of NG families such that
E0 E {H NG  H \ = } and E {H NG  H \ < }:
From these conditions, it follows easily that
__
E\ {H G  (H \ )G = ()G } = ;:
Thus, the pair (E0; E) satisfies the hypotheses of the proposition. Observe tha*
*t the
pairs (E0NG <>; ENG <>) and (E0NG <; U>; ENG <; U>) satisfy these stronger
conditions, and so satisfy the hypotheses of the proposition.
We can now prove our main approximation result.
Proof of Proposition 8.3. Observe that, by Proposition 8.4 and Remark
8.5, the canonical maps
": G nNG E(E0NG <>; ENG <>) __//_E(F0G<>; FG <>)
and
" : G nNG E(E0NG <; U>; ENG <; U>)__//E(F0G<; U>; FG <; U>)
of Definition 8.1 are Gequivalences. Let (F0; F) be a adjacent pair of Gfami*
*lies
associated to . An examination of the fixedpoint sets reveals that the map
: E(F0G<>; FG <>)__//_E(F0; F)
is a Gequivalence. The map " : G nNG E(E0NG <>; ENG <>) ___//E(F0; F)of
part (a) of the proposition is just the composite of this map and the map " of
Definition 8.1(b), which was just shown to be a Gequivalence.
For part (b) of the proposition, note that the two pairs (F0G<; U>; FG <; U>)
and (F0G<>; FG <>) satisfy the hypotheses of Lemma 6.3 so that the sequence
E(F0G<; U>; FG <; U>) __//_E(F0G<>; FG <>) __//_E(F0G<>; F0G<; U> [ FG <>)
is a Gequivariant cofibre sequence. If H 2 E0NG <; U>ENG <; U>, then H\ =
and H = H. Thus, H=H embeds as a Hspace in U. It follows that
E0NG <; U>  ENG <; U> =
{H NG  H \ = and H=H Hembeds inU}
and
F0G<; U>  FG <; U> =
{H G  (H \ )G = ()G and H=H Hembeds inU}:
The second equation above implies that the Gpair (F0G<>; F0G<; U> [ FG <>)sat
isfies the hypotheses of Proposition 6.1. Thus, for any Gspectrum D indexed on
U, the canonical map
( ^ 1) : (E(F0G<; U>; FG <; U>) ^ D)__//(E(F0G<>; FG <>) ^ D)
is a weak Gequivalence. The map (" ^ 1) of part (b) is just the composite of *
*this
map and two Gequivalences derived from the Gequivalences
: E(F0G<>; FG <>)__//_E(F0; F)
and
": G nNG E(E0NG <; U>; ENG <; U>)__//E(F0G<; U>; FG <; U>): ___
SECTION 9
Change of universe functors
and families of subgroups
In this section, we examine the question of when a Gspectrum C indexed
on a Guniverse U can be represented by a Gspectrum C0 indexed on a smaller
Guniverse U0. This question always arises when the Adams isomorphism is in
voked. In particular, the remarks preceding Theorems 2.4 and 3.3 deal with the
special cases of this question which arise in those two theorems. Section II.2*
* of
[24] addresses this issue. However, the results presented there are not as sha*
*rp
as possible, and are inadequate for our discussion of the Adams isomorphism for
incomplete universes. The essential difference between what is done here and wh*
*at
was done in Section II.2 of [24] is that here, when we compare two Guniverses U
and U0, we look at the subgroups H of G for which U and U0 are Horbit equiva
lent rather than at the subgroups H for which U and U0 are isomorphic. Thus, the
foundation for the results here is Theorem 1.2(b) of [20] rather than the earli*
*er,
weaker Corollary II.1.8 of [24]. For convenience, we assemble here a collection*
* of
relevant definitions from [20, 24].
Definition 9.1.Let U and U0 be Guniverses, and let H be a subgroup of G.
(a) The universe U embeds in U0 up to Horbits if, for each pair J K of
subgroups of H such that the orbit K=J Kembeds in U, this orbit also Kembeds
in U0. Note that, if there is a linear Hisometry from U into U0, then U embeds
in U0 up to Horbits. However, it is possible for U to embed in U0 up to Horbi*
*ts
even if there is no linear Hisometry from U into U0. For example, if G = Z=p
for some prime p, then any nontrivial Guniverse U embeds in any other nontrivi*
*al
Guniverse U0 up to Gorbits.
(b) The universes U0 and U are Horbit equivalent if, for each pair J K of
subgroups of H, the orbit K=J Kembeds in U if and only if it Kembeds in U0.
Thus, U0 and U are Horbit equivalent if and only if each embeds in the other up
to Horbits.
(c) Let "E(U; U0) "F(U; U0) be the families of subgroups of G consisting re
spectively of those H G such that U and U0 are Horbit equivalent and those
such that U embeds in U0 up to Horbits. Similarly, let E(U; U0) F(U; U0) be
the families of subgroups of G consisting respectively of those H G such that
U and U0 are Hisomorphic and those such that there is a linear Hisometry from
U into U0. Note that E(U; U0) "E(U; U0) and F(U; U0) "F(U; U0). Also observe
that, if there is a linear Gisometry i : U0 __//_U, then "E(U; U0) = "F(U; U0)*
* and
E(U; U0) = F(U; U0). The families E(U; U0) and F(U; U0) were introduced in [24]
before results like Theorem 1.2(b) of [20] were known. However, now that strong*
*er
results are available, the families E(U; U0) and F(U; U0) should be replaced by*
* the
families "E(U; U0) and "F(U; U0), respectively, in almost every context.
38
9. CHANGE OF UNIVERSE FUNCTORS AND FAMILIES OF SUBGROUPS 39
Definition 9.2.Let i : U0___//U be a linear Gisometry. A U0representation
of a Gspectrum C indexed on U is a Gspectrum C0 indexed on U0 together with
a weak Gequivalence i*C0___//C.
The main result in this section is a generalization of Theorem II.2.6 of [24*
*].
Theorem 9.3. Let i : U0___//U be a linear Gisometry, E0 and F 0be Gspectra
indexed on U0, and C be a Gspectrum indexed on U.
(a) If E0 is an "F(U; U0)spectrum, then the map
0 0 0U
i*: [E0; F 0]UG//_[i*E ; i*F ]G
is an isomorphism.
(b) If C is an "F(U; U0)spectrum, then C admits a U0representation by an
"F(U; U0)spectrum C0 indexed on U0. Moreover, C0 is unique up to Gequivalence,
and can be chosen to have cells in a canonical bijective correspondence with th*
*e cells
of C.
Proof. This result follows from Theorem 1.2(b) of [20] in the same way that*
* __
Theorem II.2.6 of [24] follows from Corollary II.1.8 of [24]. *
*__
In the setting in which this theorem is most frequently applied, is a normal
subgroup of G, G = G=, U is a Guniverse, and U0 is a trivial Guniverse which
embeds in U via a linear Gisometry i : U0___//U. Typically, in fact, U0 = U .*
* If F
is a family of subgroups of G contained in "F(U; U0) and C is a Gspectrum inde*
*xed
on U, then the spectrum EF+ ^ C is an Fspectrum and so also an "F(U; U0)
spectrum. Thus, by the theorem, there is a Gspectrum bCindexed on U0 and a
weak Gequivalence i*Cb___//EF+ ^ C. Since acts trivially on the universe U0,
bChas an associated orbit spectrum bC= which carries a canonical Gaction. This
orbit spectrum is usually denoted EF+ ^ C. Theorem 9.3 guarantees both the
existence and the naturality of this construction.
Corollary 9.4. Let be a normal subgroup of the compact Lie group G, G
be G=, U be a Guniverse, and U0 be a trivial Guniverse which Gembeds
in U. Then, for any family F of subgroups of G which is contained in "F(U; U0),
the assignment of the Gspectrum EF+ ^ C indexed on U0 to each Gspectrum C
indexed on U gives a functor from the stable category of Gspectra indexed on U*
* to
the stable category of Gspectra indexed on U0.
SECTION 10
The geometric fixedpoint functor
for incomplete universes
There are actually two definitions of the geometric fixedpoint functor . O*
*ne
is very straightforward and geometric; the other is homotopy theoretic and makes
use of universal spaces for families of subgroups and the categorical fixedpoi*
*nt
functor. The equivalence of these two rather disparate definitions is extremely
important because it allows us to use two quite different sets of techniques wh*
*en
working with this functor. Most of the basic properties of the geometric fixed*
*point
functor, including the equivalence of the two definitions, are discussed in Sec*
*tion
II.9 of [24]. That section of [24], unlike some of the other technical sections*
*, was
explicitly written so that it applies to incomplete universes. However, there *
*are
some minor oversights in what is said there. Moreover, for our work here, we ne*
*ed
a few properties of the geometric fixedpoint functor that are not covered ther*
*e. In
this section, we correct the minor difficulties in Section II.9 of [24], and su*
*pply the
necessary additional results. Once this is done, we have all that is needed for*
* the
proof of Proposition 1.2. The latter part of this section is devoted to that pr*
*oof.
We begin with the definition of in terms of universal spaces for families *
*of
subgroups and the categorical fixedpoint functor since this definition is the *
*one
most directly connected to geometric splittings.
Definition 10.1.Let be a subgroup of a compact Lie group G, U be a
Guniverse, and C be a Gspectrum indexed on U. Recall that FNG [] is the
NG family consisting of those subgroups H of NG which do not contain .
Also recall that the space E"FNG [] is the cofibre of the obvious collapse map
EFNG []+ ___//S0. The geometric fixedpoint spectrum C of C is the WG 
spectrum (E"FNG []^C) indexed on the universe U . The natural transformation
! : C __//_ C is the canonical map induced by the projection S0___//"EFNG [].
The oversights in Section II.9 of [24] must be addressed because of their im*
*pact
on the proofs of various results such as Proposition 1.2. In correcting these m*
*istakes,
we switch from the perspective of [24], which describes the geometric fixedpoi*
*nt
functor N associated to a normal subgroup N of a group G, to the perspective of
the previous sections, in which the fixedpoint functor associated to an arbi*
*trary
subgroup of G is used. Thus, the N and G of [24] are here replaced by and
NG , respectively.
Scholium 10.2. (a)Several results in Section II.9 of [24] assert the existen*
*ce
of an equivariant weak equivalence between two spectra without identifying the
map which gives that equivalence. The identity of these maps is important for o*
*ur
work. In particular, the weak WG equivalence
1 (X ) ' (1UX)
40
10. THE GEOMETRIC FIXEDPOINT FUNCTOR FOR INCOMPLETE UNIVERSES 41
of Corollary II.9.9 of [24] is the composite
1 (X ) _i_//_(1 X) ___//_(1 "EFNG [] ^ X) = (1UX);
in which i is the natural transformation introduced in Remarks II.3.14(i) of [2*
*4]
and is the map induced by the canonical projection S0 __//_"EFNG []. Moreover,
the weak WG equivalences
(E"FNG [] ^ D ^ X) ' (E"FNG [] ^ D) ^ X
and
(E"FNG [] ^ D ^ D0) ' (E"FNG [] ^ D) ^ (E"FNG [] ^ D0)
of parts (i) and (ii) of Proposition II.9.12 of [24] are the inverses of the map
(E"FNG [] ^ D) ^ X ___//_(E"FNG [] ^ D ^ X)
and the composite
(E"FNG [] ^ D) ^ (E"FNG [] ^ D0)
_!_//_(E"F " 0
NG [] ^ D ^ EFNG [] ^ D )
' (E"FNG [] ^ D ^ D0) :
Here, and ! are the natural transformations of Remarks II.3.14 (ii) and (iii) *
*of
[24], and the weak equivalence in the composite is derived from the fact that t*
*he
diagonal map
"EFNG [] __//_"EFNG [] ^ "EFNG []
is an equivariant homotopy equivalence. This identification of the maps in Coro*
*llary
II.9.9 and Proposition II.9.12 of [24] follows from an inspection of the proofs*
* of
those results and the descriptions of the natural transformations i, and ! giv*
*en
in Remarks II.3.14 of [24].
(b) Proposition II.9.13 of [24] applies for incomplete universes in spite of *
*the
fact that the proof given for it implicitly assumes that the universe is comple*
*te. In
the proof of the proposition, it is asserted that the colimit of a certain coll*
*ection of
spheres SV forms a model for the universal space "EFNG []. In fact, if the NG 
universe U is incomplete, then this colimit forms a model for the universal spa*
*ce "EF,
where F is the somewhat smaller family consisting of those subgroups K of NG
such that K does not contain and such that there is a finitedimensional subre*
*p
resentation V of U for which (V V )K 6= 0. Proposition 6.1 indicates that, fo*
*r any
NG spectrum D indexed on U, the fixedpoint spectrum (E(FNG []; F) ^ D)
is weakly WG contractible. It follows by an inspection of the cofibre sequence
E(FNG []; F) __//_"EF __//_"EFNG []
that, for the purpose of proving Proposition II.9.13, E"F may be substituted for
"EFNG [].
The first of our supplementary technical lemmas provides the basis for showi*
*ng
that, if a subgroup acts trivially on an indexing universe U, then all Gspect*
*ra
indexed on U are geometrically split at .
42 10. THE GEOMETRIC FIXEDPOINT FUNCTOR FOR INCOMPLETE UNIVERSES
Lemma 10.3. Let U be a trivial NG universe and D be an NG spectrum
indexed on U. Then the canonical map
! : D ___// D
is a weak WG equivalence.
Proof. Since there is a cofibre sequence
(EFNG []+ ^ D) __//_D _!__//_(E"FNG [] ^ D) = D;
it suffices to show that the spectrum (EFNG []+ ^D) is weakly WG contractibl*
*e.
Let F be the empty family of subgroups of NG . The space EFNG []+ may be
regarded as the universal space E(FNG []; F) of the pair (FNG []; F). Thus, by
Proposition 6.1, to show the desired weak contractibility, it suffices to show *
*that,
if H 2 FNG [], then H=H does not embed in U as a Hspace. This is obvious
since the Hisotropy subgroup of every point of the trivial universe U_must_
contain and so cannot be H. __
Our second supplementary technical lemma describes the interaction of change
of universe functors and geometric fixedpoint functors. It is the key to showi*
*ng that
change of universe functors preserve geometric splittings. Since geometric fix*
*ed
point functors for two different universes appear in this result, the notation *
*U is
used here to denote the geometric fixedpoint functor associated to the univers*
*e U.
Lemma 10.4. Let i : U __//_U0be a NG isometry between two NG universes
U and U0, and i : U ___//(U0)be the isometry obtained from i by passage to
fixedpoints. If D is an NG spectrum indexed on U, then there are natural
WG maps
ffi : i*(D )___//(i*D)
and
^ffi: i*(U D) __//_U0(i*D)
making the diagram
i*(D )___ffi_//(i*D)
i*!  !
fflffl^ffi fflffl
i*(U D) _____//U0(i*D)
commute. Moreover, the map ^ffiis a weak WG equivalence.
Proof. Let j : U ___//U and k : (U0) ___//U0be the inclusions of the two
fixed universes. To keep track of the change of universe functors appearing in
this proof, we expand our usual notation for fixedpoint spectra to include exp*
*licitly
the change of universes involved in the passage to fixed points. Thus, the spec*
*trum
usually denoted D is here denoted (j*D) .
The natural map ffi is a composite of the form
i*((j*D) ) __//_(i*j*D) __//_(k*i*D) :
The first map in this composite is an instance of a WG map i*(E ) __//_(i*E)
applicable to all NG spectra E indexed on U . This natural map is the adjoint
of the NG map i*(E ) __//_i*E derived from the inclusion E __//_E. The second
10. THE GEOMETRIC FIXEDPOINT FUNCTOR FOR INCOMPLETE UNIVERSES 43
map in the composite is obtained by passage to fixed points from the natural
NG map i*j*D __//_k*i*D which is the adjoint of the map
k*i*j*D ~=i*j* j*D _i*ffl//_i*D:
Here, ffl is the counit of the (j* ; j*)adjunction and the isomorphism is deri*
*ved from
the equation k O i = i O j . The natural map ^ffiis just the composite
i*U D = i*((j*(E"FNG [] ^ D)) ) _ffi//_(k*i*(E"FNG [] ^ D)) = U0(i*D):
The diagram asserted to commute is therefore just a naturality diagram for ffi.
In order to show that ^ffiis a weak WG equivalence, we must use the alterna*
*tive
geometric definition of the geometric fixedpoint functor which is given in Def*
*inition
II.9.7 of [24]. To distinguish the geometric version of the functor from the t*
*he
homotopytheoretic version, we denote the geometric version from Definition II.*
*9.7
of [24] by "U . This functor is applicable to NG spectra indexed on U. It is
constructed in [24] by first introducing a prespectrumlevel version and then p*
*assing
to the spectrumlevel version using an approximation by CWprespectra. It is ea*
*sy
to see that, if we apply the prespectrumlevel versions of the composite functo*
*rs
i*OU" and "U0Oi* to a inclusion prespectrum "Dindexed on U, then the resulting
prespectra i*("U "D) and "U0(i*D") are naturally isomorphic. It follows that th*
*ere
is a natural weak WG equivalence
"ffi: i*U" D___//"U0(i*D)
relating the spectrumlevel functors. Under the natural weak WG equivalence
: U D = (j*(E"FNG [] ^ D)) __//_"UD
of Theorem II.9.8(ii) of [24] and the analogous map for the universe U0, the na*
*tural
map ^ffiis identified with "ffiand so is a weak WG equivalence. *
* ___
We conclude this section with the proof of Proposition 1.2.
Proof of Proposition 1.2. Part (a) of the proposition follows directly from
Corollary II.9.9 of [24] and Scholium 10.2, which identifies the map asserted t*
*o be
a weak equivalence in the corollory. Parts (b) and (c) follow directly from Lem*
*mas
10.3 and 10.4, respectively. Scholium 10.2 and Proposition II.9.12 of [24] prov*
*ide all
that is needed for the proofs of parts (d) and (e). Part (f) follows from an ob*
*vious
diagram chase. Part (g) follows from parts (b), (c), and (e), since the localiz*
*ation
CT of C at T can be identified with C ^ jG*S0T, where S0Tis the localization at*
* T
of the sphere spectrum S0 indexed on the trivial Guniverse UG . Part (h) follo*
*ws
from Proposition 3.1 of [7], which describes the interaction between completion*
*s_
and the geometric fixedpoint functor. __
SECTION 11
The Wirthm"uller isomorphism
for incomplete universes
Let N be a subgroup of a compact Lie group G, L be the Nrepresentation
given by the tangent space of G=N at the identity coset eN, and D be a Nspectr*
*um
indexed on a complete Guniverse. Then the Wirthm"uller isomorphism (see [29]
and Theorem II.6.2 of [24]) identifies the Gspectra G nN D and FN [G; LD). It
is easy to see that the existence of such an isomorphism for every Nspectrum D
implies that the Gspectrum 1 G=N+ has a wellbehaved SpanierWhitehead dual.
This observation severely restricts the extent to which the Wirthm"uller isomor*
*phism
can be extended to an incomplete Guniverse U since, by Proposition 7.1, the G
spectrum 1UG=N+ cannot have a wellbehaved SpanierWhitehead dual unless the
orbit G=N embeds as a Gspace in U.
What can be expected for an incomplete universe is a pair of partial extensi*
*ons.
One result in this pair should assert that, if the orbit G=N does embed, then t*
*here
is an Wirthm"uller isomorphism for Nspectra indexed on U just as in the context
of a complete universe. The other result should indicate that, even if the orb*
*it
does not embed, there is some reasonable Nfamily W of subgroups such that a
Wirthm"uller isomorphism exists for all Wspectra indexed on U. The first resul*
*t of
this pair is contained implicitly in Section II.6 of [24], and is explicitly pr*
*ovided as
the first result in this section. The present status of the second part of this*
* pair is
less satisfactory. For trivial reasons, there is a family W(N; G; U) of subgrou*
*ps of
N such that a Wirthm"uller isomorphism exists for all W(N; G; U)spectra indexed
on U. This family is maximal in the sense that, for any Nfamily F not contained
in W(N; G; U), there is an Fspectrum D for which there is no Wirthm"uller isom*
*or
phism. However, for an arbitrary compact Lie group G, very little is known about
the family W(N; G; U) beyond these formal properties. Thus, here we introduce
two more tractable families of subgroups, W0(N; G; U) and W00(N; G; U), and sta*
*te
several results about them which are first approximations to the desired second
half of our pair. In particular, we show that, if G is a finite group, then the*
* mys
terious family W(N; G; U) is just the more accessibly defined family W0(N; G; U*
*).
The family W00(N; G; U) is the most easily understood of the three. Moreover,
for any compact Lie group G, any subgroup N, and any Guniverse U, there is a
Wirthm"uller isomorphism for all W00(N; G; U)spectra indexed on U.
In the places where a Wirthm"uller isomorphism is used in the proofs of our
splitting theorems, the subgroup N is the normalizer of another subgroup . In t*
*his
situation, our results on the Wirthm"uller isomorphism take a particularly simp*
*le
form. In addition to playing a role in the proofs of our splitting theorems, t*
*he
results presented here provide insight into the conditions under which Theorem *
*3.8
can be extended to a result more like Theorems 2.4 and 3.3.
44
11. THE WIRTHM"ULLER ISOMORPHISM FOR INCOMPLETE UNIVERSES 45
The approach taken to the Wirthm"uller isomorphism in Theorem II.6.2 of [24]
is not suitable in the context of incomplete universes because the map
! : FN [G; LD) __//_G nN D;
of that theorem is not even defined if the orbit G=N does not embed in U as a
Gspace. There is, however, another comparison map
: G nN D ___//FN [G; LD);
introduced in Definition II.6.8 of [24], which exists for any Guniverse U. The
proper approach to the Wirthm"uller isomorphism for incomplete universes is to
look for conditions under which this map is an isomorphism. Hereafter, we ref*
*er
to as the Wirthm"uller map.
If G=N embeds in U as a Gspace so that the map ! exists, then results
contained in [24] suffice to show that and ! are inverse Gequivalences.
Proposition 11.1. Let N be a subgroup of a compact Lie group G, U be a
Guniverse into which G=N embeds as a Gspace, and D be a NCW spectrum
indexed on U. Then the maps
: G nN D ___//FN [G; LD)
and
! : FN [G; LD) __//_G nN D
are inverse weak Gequivalences.
Proof. Let V be a finite dimensional Grepresentation contained in U into
which G=N embeds as a Gspace. As a Nrepresentation, V decomposes as a direct
sum of L and some other Nrepresentation W . Thus, L is contained in U as a N
representation. By Theorem I.6.1 of [24], suspension by L is an invertible func*
*tor
on the equivariant stable category of Nspectra indexed on U. Also, suspension *
*by
V is an invertible functor on the equivariant stable category of Gspectra inde*
*xed
on U. Since the invertibility of suspension by L and V are the only two propert*
*ies
of the equivariant stable category used in the proof of the Wirthm"uller isomor*
*phism__
in section II.6 of [24], that proof extends to a proof of this proposition. *
* __
To begin our discussion of the availability of the Wirthm"uller isomorphism *
*when
the orbit G=N does not embed in U, we define the three families of subgroups
of N mentioned in the introduction. The kindest thing that can be said about
the definitions of these families is that they are less enlightening than one w*
*ould
like. This section concludes with an investigation of a very special case of t*
*he
Wirthm"uller isomorphism. This special case offers some insight into the stran*
*ge
nature of our three families of subgroups.
Definition 11.2.Let N be a subgroup of a compact Lie group G, and let U
be a Guniverse.
(a) Let W(N; G; U) be the family of subgroups K of N such that, for every
Kspectrum C indexed on U, the map
: G nN (N nK C) ___//FN [G; LN nK C)
is a weak Gequivalence.
46 11. THE WIRTHM"ULLER ISOMORPHISM FOR INCOMPLETE UNIVERSES
(b) Let W0(N; G; U) be the family of subgroups K of N such that, for every
pair of subgroups J of K and H of G with J H, the orbit (N \ H)=J embeds as
an (N \ H)space in U if and only if the orbit H=J embeds as a Hspace in U.
(c) Let W00(N; G; U) be the family of subgroups K of N for which there is a
subgroup H of N such that K H and G=H embeds as a Gspace in U.
Note that, for any family W of subgroups of N, to show that the Wirthm"uller
map is a weak Gequivalence for every Wspectrum D indexed on U, it suffices
to show that the map
: G nN 1UN=K+ ___//FN [G; L1UN=K+ )
is a weak Gequivalence for every K 2 W. With this observation, it is easy to s*
*ee
that W(N; G; U) is maximal among the families of subgroups of N for which there
is a Wirthm"uller isomorphism.
Proposition 11.3. Let N be a subgroup of a compact Lie group G, and let U
be a Guniverse. Then, for every W(N; G; U)spectrum D indexed on U, the map
: G nN D ___//FN [G; LD);
is a weak Gequivalence. Moreover, if F is any family of subgroups of N which
is not contained in W(N; G; U), then there is an Fspectrum D0 indexed on U for
which the Wirthm"uller map is not an isomorphism.
This result would be somewhat vacuous without the following theorem and
conjecture.
Theorem 11.4. Let N be a subgroup of a finite group G, and let U be a G
universe. Then, W(N; G; U) = W0(N; G; U). Thus, for any W0(N; G; U)spectrum
D indexed on U, the natural map
: G nN D ___//FN [G; LD);
is a weak Gequivalence.
Conjecture 11.5. Let N be a subgroup of a compact Lie group G, and let U
be a Guniverse. Then, W(N; G; U) = W0(N; G; U).
The proof of the theorem above, which is given in Section 15, makes use of
the fact that, if G is a finite group and N and H are subgroups, then the orbit
G=N, regarded as a Hspace, decomposes into a disjoint union of Horbits. The
obvious way to try to extend this proof to a proof of the conjecture would be to
replace this decomposition by the skeletal filtration of G=N regarded as a HCW
complex. This approach cannot work because the orbit G=N (or, more precisely,
some closely related constructions) appears covariantly in the domain of the map
and contravariantly in the range. It seems likely that a rather delicate geom*
*etric
argument will be needed to prove the conjecture.
The utility of Proposition 11.3 and Theorem 11.4 is limited by the fact that
the families of subgroups associated to those results are not easily understood*
*. The
family W00(N; G; U) is typically smaller than either of these, but is certainly*
* much
easier to work with. Thus, the following version of the Wirthm"uller isomorphism
theorem is likely to be the most general one which is widely applicable in the *
*context
of incomplete universes.
11. THE WIRTHM"ULLER ISOMORPHISM FOR INCOMPLETE UNIVERSES 47
Theorem 11.6. Let N be a subgroup of a compact Lie group G, and let U be
a Guniverse. Then, for every W00(N; G; U)spectrum D indexed on U, the map
: G nN D ___//FN [G; LD);
is a weak Gequivalence.
Proof. As we have already noted, it suffices to consider the case in which
D = 1UN=K+ , for K 2 W00(N; G; U). Throughout this proof, we work with
the suspension spectra of orbits. Thus, to reduce notational clutter, we omit *
*all
instances of 1Ufrom our notation. Select a subgroup H of G such that K H N
and G=H embeds in U as a Gspace. Denote the Nrepresentation given by the
tangent space of G=N at the identity coset eN by L(N) rather than just L. Also,
denote the Hrepresentations given by the tangent spaces of N=H and G=H at
the identity cosets eH by L(H; N) and L(H), respectively. As a Hrepresentation,
L(H) is the direct sum of L(H; N) and L(N).
By Lemma II.6.13 of [24], the diagram
G nN N=K+ _______________//FN [G; L(N)N=K+ )
~= ~=
fflffl fflffl
G nH H=K+ FN [G; L(N)N nH H=K+ )
~= ~=FN[G;L(N))
fflffl ~= fflffl
FH [G; L(H)H=K+ ) ____//_FN [G; L(N)FH [N; L(H;N)H=K+ ))
commutes in the equivariant stable category of Gspectra indexed on U. The
two unlabeled vertical isomorphisms are derived from the spacelevel isomorphism
N=K+ ~=N nH H=K+ and one of the isomorphisms given by Lemma II.4.10 of [24].
The horizontal isomorphism at the bottom of the diagram is also given by Lemma
II.4.10 of [24]. The vertical maps labeled and FN [G; L ) are isomorphisms by
Proposition 11.1 since the orbit G=H embeds in U as a Gspace. It now follows
that the upper horizontal arrow
: G nN N=K+ ___//FN [G; L(N)N=K+ )
is an isomorphism in the equivariant stable category of Gspectra indexed on U.*
* ___
Remark 11.7. Theorem 11.6 implies, of course, that the family W00(N; G; U) *
*is
contained in the Wirthm"uller family W(N; G; U). It is fairly easy to see that,*
* as the
conjecture would lead us to expect, W00(N; G; U) is also contained in W0(N; G; *
*U).
In the instances where the Wirthm"uller isomorphism is needed for the proof *
*of
our splitting theorems, the subgroup which has so far been denoted N is actuall*
*y the
normalizer NG of some other subgroup . In this context, a connection between
the family E0NG <; U> of Definition 8.1(c) and the family W00(NG ; G; U) makes *
*it
easier to verify that certain spectra satisfy the hypotheses of Theorem 11.6. A*
*s in
our earlier notation, the NG representation L is just that derived from the ta*
*ngent
space of G=NG at the identity coset eNG .
Theorem 11.8. Let be a normal subgroup of a compact Lie group G,
be a subgroup of , and U be a Guniverse. Then the NG family E0NG <; U>
48 11. THE WIRTHM"ULLER ISOMORPHISM FOR INCOMPLETE UNIVERSES
is contained in the NG family W00(NG ; G; U). Thus, if D is a E0NG <; U>
spectrum indexed on U, then the natural map
: G nNG D ___//FNG [G; LD)
is a weak Gequivalence. In particular, for any NG spectrum C indexed on U, the
maps
: G nNG (EE0NG <; U>+ ^ C) __//_FNG [G; LEE0NG <; U>+ ^ C)
and
: G nNG (E(E0NG <; U>; ENG <; U>) ^ C) __//_
FNG [G; LE(E0NG <; U>; ENG <; U>) ^ C)
are weak Gequivalences.
This result follows from a simple technical lemma about the isotropy subgrou*
*ps
of arbitrary Gsets.
Lemma 11.9. Let be a normal subgroup of the group G, Q be a subgroup of
G, X be a Gset, and x be a point of X such that the Qisotropy subgroup (Q)x
of x is Q. Then the Gisotropy subgroup Gx of x is contained in the Gnormalizer
of Q \ .
Proof. Let g 2 G such that gx = x. Since Q and (Q)x = Q,
x = Q\ and (gQg1)gx = gQg1. The subgroup gQg1 is equal to gQg1
since is normal. Thus,
gx = (gQg1)gx\ = gQg1 \ :
The normality of also implies that gQg1 \ = g(Q \ )g1. The equality
gx = x then implies the sequence of identifications
Q \ = x = gx = gQg1 \ = g(Q \ )g1:
It follows that g is in the Gnormalizer of Q \ . __*
*_
Proof of Theorem 11.8. Let K be in E0NG <; U>. Then K \
and K=K Kembeds in U. Clearly K \ is . Applying the lemma
with the subgroup Q of the lemma taken to be K provides a subgroup H of
G such that K H NG and G=H embeds as a Gspace in U. Since
K K H, K is in W00(NG ; G; U). The remaining assertions of the theo
rem follow from Theorem 11.6 and the fact that the spectra EE0NG <; U>+ ^_C_
and E(E0NG <; U>; ENG <; U>) ^ Care both E0NG <; U>spectra. __
The remainder of this section is devoted to an examination of a very spe
cial case of the Wirthm"uller isomorphism. This investigation leads incidental*
*ly
to a proof that, for G finite, the family W(N; G; U) is contained in the family
W0(N; G; U). However, its deeper purpose is to provide the intuition behind the*
* def
inition of W0(N; G; U) and to give some insight into the conditions needed to o*
*btain
a Wirthm"uller isomorphism. For this investigation, assume that G is a finite g*
*roup
and that K is a subgroup of N contained in W(N; G; U). Since K 2 W(N; G; U),
the composite
1UG=K+ ~=G nN 1UN=K+ ___//_FN [G; 1UN=K+ )
11. THE WIRTHM"ULLER ISOMORPHISM FOR INCOMPLETE UNIVERSES 49
is a weak Gequivalence. Thus, for any subgroup H of G, the composite
[1UG=H+ ; 1UG=K+ ] UG __//_[1UG=H+ ; FN [G; 1UN=K+ )]UG
~= [1UG=H+ ; 1UN=K+ )] UN
is an isomorphism.
The domain and range of this isomorphism can be computed using Corollary
5.3(b), which asserts that each of them is a free abelian group. The generators*
* of
[1UG=H+ ; 1UG=K+ ] UGare equivalence classes of certain diagrams of the form
G=H oof_G=J _p_//_G=K;
where J K, p is the obvious projection, and f is a Gmap between Gsets. If the
map f takes the coset eJ to the coset gH, then g1Jg H. The allowed diagrams
are those for which H=(g1Jg) embeds in U as an Hspace. Analogously, the
generators of [1UG=H+ ; 1UN=K+ )] UNare equivalence classes of certain diagrams
of the form
G=H ooh_N=J _q_//N=K;
where J K, q is the obvious projection, and h is an Nmap between Nsets. If h
takes the coset eJ to the coset gH, then g1Jg g1Ng\H. The allowed diagrams
are those for which (g1Ng \ H)=(g1Jg) embeds in U as an (g1Ng \ H)space.
The inverse of the Wirthm"uller isomorphism displayed above takes the gener
ator associated to the diagram
G=H ooh_N=J _q_//N=K
to the generator associated to the diagram
"h 1nNq
G=H oo__G nN N=J ______//_G nN N=K ~=G=K;
where "his the Gmap induced by the Nmap h. From the existence of a Wirthm"ull*
*er
isomorphism for the Nspectrum 1UN=mgK+ , we can therefore derive the condi
tion that, for every pair of subgroups J of N and H of G and every element g of*
* G
such that g1Jg g1Ng \ H, the orbit (g1Ng \ H)=(g1Jg) embeds in U as an
(g1Ng \ H)space if and only if the orbit H=(g1Jg) embeds in U as an Hspace.
In fact, it is easy to check that this embedding condition for arbitrary g 2 G *
*fol
lows from the analogous condition in which g is restricted to being e. But this*
* is
precisely the condition that K be in W0(N; G; U). Thus, we have shown that, if G
is finite, then W(N; G; U) is contained in W0(N; G; U). If G is not finite, the*
*n the
Gorbit G=H, considered as an Nspace, need not decompose into a disjoint union
of orbits. Thus, Corollary 5.3(b) cannot be applied, and the argument presented
here does not suffice to show the containment.
SECTION 12
An introduction to the Adams isomorphism
for incomplete universes
One of the ways in which Theorem 3.8 differs from Theorems 2.4 and 3.3 is
that the summand of the splitting indexed on a subgroup of is described
in terms of the WG spectrum E(; ; G; U)+ ^ C[] rather than in terms of a
WG =W spectrum of the form E(; ; G; U)+ ^W Ad(W ) C[]. This differ
ence is necessary because the Adams isomorphism (see section 5 of [1] and secti*
*on
II.7 of [24]) need not be available if the universe U is incomplete. Here we d*
*is
cuss the conditions under which an Adams isomorphism is available for incomplete
universes.
There are two easily seen obstacles to obtaining an Adams isomorphism of
the desired sort. The first is that any sort of Adams isomorphism must use a
representability theorem like Theorem II.2.6 of [24] to approximate the spectrum
E(; ; G; U)+ ^ C[] indexed on the universe U by a spectrum indexed on some
W trivial universe so that an orbit spectrum can be constructed. If the G
universe U is complete, then such an approximation is certainly available. How
ever, if U is incomplete, then the approximation need not exist, and, even if it
does, Theorem II.2.6 of [24] may be inadequate to imply its existence. Moreover,
the approximation problem is exacerbated by the fact that the natural indexing
universe for such an approximating spectrum is (U )W = UN , whereas, for
the purposes of our splitting theorem, the only reasonable indexing universe is*
* U .
If U is complete, then U and UN are equivalent WG universes, so this appar
ent difference in indexing universe is immaterial. However, if U is not comple*
*te,
then there is no reason to assume that U and UN are equivalent as WG 
universes. Thus, the kind of approximation needed for an incomplete universe is
more farreaching than the one needed for a complete universe. This approximati*
*on
problem is, in fact, resolved by Theorem 9.3 and Lemma 12.3, which provide the
desired approximation under any conditions where other, more serious, problems
do not eliminate the possibility of an Adams isomorphism.
In the context of a complete universe, the W freeness of a spectrum like
E(; ; G; U)+ ^ C[] suffices to guarantee the existence of an Adams isomor
phism. However, in the context of an incomplete universe, this condition need n*
*ot
be sufficient. Thus, the more serious obstacle to obtaining the desired isomorp*
*hism
is the necessity of identifying conditions under which it can be expected to ex*
*ist.
These conditions could take the form of rather stringent constraints on the uni
verse U or of some new restrictions on the spectra to which we wish to apply the
isomorphism.
To describe our resolution of this more serious problem, it seems best to si*
*m
plify our notation. In the context of our splitting theorems, we want an Adams
isomorphism for a certain W free WG spectrum indexed on U . However, for
50
12. THE ADAMS ISOMORPHISM FOR INCOMPLETE UNIVERSES 51
our discussion of the existence of this isomorphism, what matters about these t*
*wo
groups is that W is a normal subgroup of WG . Thus, for the rest of our dis
cussion of the Adams isomorphism, we replace WG by G, W by , and the
WG universe U by the Guniverse U. We are then looking for an Adams isomor
phism for a free Gspectrum D. As in [24], we denote the adjoint representation
of G derived from by A. The differences between this notation and the notation
in Section II.7 of [24] are that we are using instead of N for the normal subg*
*roup
and G rather than J for the quotient group G=.
To deal with the difficulty that we wish to index the orbit spectrum D= on a
universe other than the natural one, we replace the universe U that is the ind*
*exing
universe of D in [24] with an arbitrary trivial Guniverse U0. We also replace
the inclusion i of U into U by an arbitrary linear Gisometry i : U0___//U. Th*
*us,
in the remainder of this section, the spectrum D, all of our fixedpoint spect*
*ra,
and all of our orbit spectra are indexed on U0 rather than U .
With this notation in place, we can describe another problem with the Adams
isomorphism that arises in incomplete universes. In [24], the Adams isomorphism
is derived from a dimensionshifting transfer of the form o : i*(D=) ___//A i**
*D.
The desuspension by A appearing here may be undefined if A is not isomorphic to*
* a
subrepresentation of U. This difficulty can be avoided by working with a transf*
*er of
the form o : i*((A D)=) __//_i*D. Such a change amounts to nothing more than
replacing D by A D in Theorem II.7.1 of [24]. In the applications of the Adams
isomorphism to our splitting theorems, this replacement is, in fact, always mad*
*e.
Thus, for our purposes, an Adams isomorphism derived from this second type of
transfer is preferred. Hereafter, a dimensionshifting transfer of the form
o : i*((A D)=) ___//i*D
is referred to as an Adams transfer.
We deal with the problem that the freeness of the action on D is not a
sufficient condition for the existence of the desired Adams isomorphism by impo*
*sing
on the universe U the minimal conditions under which an Adams isomorphism
might possibly exist and then describing the additional conditions that must be
imposed on D to obtain the isomorphism. The minimal condition on U is that the
free orbit =e embeds in U as a space. The conditions that must be imposed on
D depend on both U and U0, but not the choice of i. They are best described in
terms of a family AG (; U; U0) of subgroups of G. This family is one of two nat*
*ural
generalizations to incomplete universes of the family FG () of Definitions II.2*
*.3(ii)
of [24].
Definition 12.1.If is a normal subgroup of a compact Lie group G, U is a
Guniverse into which =e embeds as a space, and U0 is a trivial Guniverse
that embeds in U, then the family AG (; U; U0) of subgroups of G consists of th*
*ose
H G such that
(i)H \ = {e}
(ii)H=H embeds in U as an Hspace
(iii)for every subgroup K of H and Q of G such that K Q, the orbit Q=K
embeds in U as a Qspace if and only if the orbit Q=K embeds in U0 as a
Qspace.
If U is a complete Guniverse and U0 = U , then AG (; U; U0) = FG (). For any
universe U, we denote AG (; U; U ) by AG (; U). If = G, then, for any trivial
52 12. THE ADAMS ISOMORPHISM FOR INCOMPLETE UNIVERSES
universe U0, AG (; U; U0) is just the family FG (G) consisting of the trivial s*
*ubgroup
of G.
Our generalization of Theorem II.7.1 of [24] to incomplete universes can now
be stated. This result is proven in sections 16 and 17.
Theorem 12.2. Let be a normal subgroup of a compact Lie group G, G be
G=, A be the adjoint representation of G derived from , U be a Guniverse into
which =e embeds as a space, U0 be a trivial Guniverse, and i : U0___//U be
a linear Gisometry. If D is an AG (; U; U0)spectrum indexed on U0, then there
is an Adams transfer
o : i*((A D)=) ___//i*D
whose adjoint
o": (A D)= ___//(i*i*D)
is a weak Gequivalence of Gspectra indexed on U0. Thus, for any Gspectrum B
indexed on U0, there is a natural isomorphism
A U0 "o* * U0 U
B; ( D)= G __~=_//B; (i i*D)G ~=[i*B; i*D]G:
In order to apply this theorem to our splitting results, we must deal with
the representability problem mentioned in the introduction to this section. The
issue is that Theorem 12.2 applies to AG (; U; U0)spectra indexed on U0, but, *
*in
practice, one always begins with an AG (; U; U0)spectrum C indexed on U. To
apply the theorem, one must find an AG (; U; U0)spectrum D indexed on U0 such
that i*D ' C. As in the remarks preceding Theorems 2.4 and 3.3 and Corollary
9.4, we denote the spectrum (A D)= indexed on U0 by (A C)=. The Adams
isomorphism then becomes a weak Gequivalence "o: (A C)= __//_(i*C). The
following result indicates that, if C is an AG (; U; U0)spectrum, then the des*
*ired
spectrum D can always be found. For this result, recall from Definition 9.1 the
family "E(U; U0) of subgroups associated to the universes U and U0 and the noti*
*on
of two universes being Gorbit equivalent.
Lemma 12.3. Let be a normal subgroup of a compact Lie group G, U be a
Guniverse into which =e embeds as a space, U0 be a trivial Guniverse, and
i : U0___//U be a linear Gisometry. If H 2 AG (; U; U0), then the universes U *
*and
U0 are Horbit equivalent. Thus, the family AG (; U; U0) is contained in the fa*
*mily
"E(U; U0).
Proof. Let H 2 AG (; U; U0). Since U0 embeds in U as a Guniverse, U0 is
clearly contained in U up to Horbits. Assume that J K H and that K=J
embeds as a Kspace in U. We must show that it also embeds in U0. Because K H
and H 2 AG (; U; U0), K=K embeds as a Kspace in U. But then K=J must
also embed as a Kspace in U because the set of Kisotropy subgroups of U
is closed under intersection. By condition (iii) of the definition of AG (; U;*
* U0),
K=J must then embed as a Kspace in U0. Since K \ = {e}, K=J embeds __
as a Kspace in U0, and U is contained in U0 up to Horbits. _*
*_
There are two ways in which Theorem 12.2 might seem unsatisfactory. Cer
tainly, the family AG (; U; U0) can be rather smaller than we would like. Also,*
* the
motivation behind its definition is hardly obvious. The remainder of this secti*
*on
12. THE ADAMS ISOMORPHISM FOR INCOMPLETE UNIVERSES 53
is devoted to providing the intuition behind this definition; it contains a heu*
*ristic
argument that the hypotheses of the theorem are necessary as well as sufficient.
The purpose of condition (i) in the definition should be clear; it ensures that
AG (; U; U0)spectra are free. Without such a freeness condition, one cannot
hope to have the Adams transfer from which the Adams isomorphism is obtained.
The other two conditions are more exotic looking. However, examining the Adams
isomorphism in a very special case suffices to explain the need for conditions *
*(ii)
and (iii). For the remainder of this section, assume that G is a finite group a*
*nd H
is a subgroup of G such that H \ = {e}. Then G=H is a free Gspace. Note
that (G=H)= is just G=H. If there were an Adams isomorphism theorem for
the spectrum D = 1U0G=H+ , then its represented form would assert that, for each
subgroup Q of G such that Q, the composite
0 1 1 U0
[1U0G=Q+ ; 1U0G=H+ ]UG ~= [U0G=Q+ ; U0G=H+ ] G
_i*_// [1 1 U
U G=Q+ ; U G=H+ ] G
_o*_// [1 1 U
U G=Q+ ; U G=H+ ]G
is an isomorphism, where o : 1UG=H+ ___//1UG=H+ is the standard transfer
associated to the projection G=H __//_G=H. In order for the transfer o to exist
as a map between spectra indexed on U, the orbit H=H must embed in U as an
Hspace. Thus, condition (ii) in the definition is necessary for the existence*
* of
the map which is supposed to give the Adams isomorphism.
In order to understand the third condition in the definition, let us assume *
*that
H satisfies the first two conditions so that the composite above is welldefine*
*d, and
let us consider the question of whether this composite is an isomorphism. This
question can be addressed by using Corollary05.3(b) to take a closer look at the
morphism sets [1U0G=Q+ ; 1U0G=H+ ]UG and [1UG=Q+ ; 1UG=H+ ]UG. This theo
rem indicates that [1UG=Q+ ; 1UG=H+ ]UGis a free abelian group whose generators
are equivalence classes of certain diagrams of the form
G=Q oop_G=K _f_//_G=H;
where K Q, p is the obvious projection, and f is a Gmap between Gsets.
The allowed diagrams are those for which0Q=K embeds in U as a Qspace. The
morphism set [1U0G=Q+ ; 1U0G=H+ ] UGhas an analogous description; each of its
generators can be represented by a diagram of the form
0 h
G=Q oop_G=K ___//_G=H
in which K is a subgroup of G subconjugate to H such that K Q. Again, p0is
the obvious projection and h is a Gmap.
The image of the generator associated to the diagram (p0; h) under the first
two maps in the composite above is just the generator of [1UG=Q+ ; 1UG=H+ ] UG
represented by the same diagram. The transfer o which induces the third map in
the composite is represented by the diagram
G=H oot_G=H _id//_G=H;
54 12. THE ADAMS ISOMORPHISM FOR INCOMPLETE UNIVERSES
where t is just the standard projection. The composite of o with our generator *
*in
[1UG=Q+ ; 1UG=H+ ] UGcan therefore be computed from the diagram
G=K H
q uuuu HHHkH
uu HHH
zzuuu H$$
G=K J G=H F
p0vvvv JJJhJ tvvvv FFidF
vv JJJ vvv FFF
vvv J$$ zzvv F##
G=Q G=H G=H
of Gsets and Gmaps in which the quadrilateral is a pullback and q is the stan*
*dard
projection. Thus, the image of a generator
0 h
G=Q oop_G=K ___//_G=H
0
of [1U0G=Q+ ; 1U0G=H+ ] UGunder our putative Adams isomorphism is the mor
phism represented by the diagram
G=Q oop_G=K _k_//_G=H:
Note that the diagram
0 h
G=Q oop_G=K ___//_G=H
0
represents an actual generator of [1U0G=Q+ ; 1U0G=H+ ]UG only when Q=K em
beds as a Qspace in U0. Similarly, the diagram
G=Q oop_G=K _k_//_G=H
represents an actual generator of [1UG=Q+ ; 1UG=H+ ] UGonly when Q=K embeds
in U as a Qspace. Thus, condition (iii) of the definition is essentially the c*
*ondition
H must satisfy to ensure that the composite above is an isomorphism. Our analys*
*is
suggests that condition (iii) should be imposed on subgroups K of G which are
subconjugate to H. However, it is easy to check that it suffices to consider t*
*he
actual subgroups K of H. This discussion indicates that, at least when G is fin*
*ite,
the family AG (; U; U0) is the largest family for which we can hope to have an
Adams isomorphism.
Part 3
The Longer Proofs
SECTION 13
The proof of Proposition 3.10
and its consequences
The first purpose of this section is to provide the proofs of Proposition 3.*
*10 and
two related results, Propositions 2.8 and 3.6. These proofs provide the motivat*
*ion
for the definitions of two maps which play a critical role in the proofs of our*
* main
splitting theorems. The second purpose of this section is to introduce these ma*
*ps,
and to use them to prove a special case of Theorem 3.2.
Proof of Proposition 3.10. Assume that C is a Gspectrum indexed on a
Guniverse U, is a normal subgroup of G, G = G=, and B is a Gspectrum
indexed on U . Also, let be a subgroup of , and (F0; F) be a adjacent
pair of families of subgroups of G associated to . Let L be the tangent NG 
representation at the identity coset eNG of G=NG . Note that, by Lemma
V.10.3(ii) of [24], L = 0. The pair (E0NG <; U>; ENG <; U>) of families of su*
*b
groups of NG (see Definition 8.1(c)) appears throughout this proof, and, to re*
*duce
notational clutter, we denote this pair by (E', E) hereafter in the proof. For *
*the
same reason, the quotient group WG =W is denoted here by W.
The isomorphism fl of Proposition 3.10 is the composite:
0 U
j* B; E(F ; F) ^GC
__(("^1)1)*//_ 0 U
~= j* B; G nNG E(E ; E) ^GC
_(i1)*//_ 0 U
~= j* B; G nNG (E(E ; E) ^ C)G
" L 0 U
_~=_// j* B; E(E ; E) ^ CNG
_~=//_ ; L 0 U
j* B; ( E(E ; E) ^ C) WG
__*__// h ; L 0 " iU
~= j* B; ( E(E ; E) ^ EFNG [] ^ C) WG
_("1)*//_ h ; " Ui
~= j* B; E(; ; G; U)+ ^ (E FNG [] ^ C)WG
; U
= j* B; E(; ; G; U)+ ^ C WG :
Here, ((" ^ 1)1)* is the inverse of the isomorphism given by Proposition 8.3. *
*The
isomorphism (i1)* is derived from the inverse of the isomorphism i of Lemma
II.4.9 of [24], and the isomorphism " is derived from the Wirthm"uller isomorph*
*ism
of Theorem 11.8. The unlabeled isomorphism is the adjunction isomorphism for
56
13. THE PROOF OF PROPOSITION 3.10 57
the spectrumlevel fixedpoint functor. The projection
: E(E0; E)__//E(E0; E) ^ "EFNG []
is easily seen to be a weak NG equivalence by checking fixedpoint sets. Thus,
the map * in the composite above is an isomorphism. The isomorphism ("1)*
is derived from the map of Remarks II.3.14(ii) of [24] via the identification *
*in
Lemma 8.2 of the fixedpoint set of E(E0; E). Scholium 10.2(a) and Proposition
II.9.12(i) of [24] indicate that the map is an isomorphism in this case.
The weak Gequivalence O of Proposition 3.10 involves several isomorphisms
relating various change of group and change of universe functors. Thus, in desc*
*rib
ing this weak equivalence, we temporarily abandon the convention of suppressing
from our notation the change of universe functor included in any spectrumlevel
fixedpoint functor. The weak equivalence of the proposition is then the compos*
*ite:
(j*(E(F0; F) ^ C))
_(("^1)_)1//_ * 0
' (j ((G nNG E(E ; E)) ^ C))
__(i_)1//_ * 0
~= (j G nNG (E(E ; E) ^ C))
____// * L 0
' (j FNG [G; E(E ; E) ^ C))
_~=_// * L 0
(FNG [G; j ( E(E ; E) ^ C)))
_~=_// * L 0 W
FW [G; ((j ( E(E ; E) ^ C)) ) )
_~=_// * * L 0 W
FW [G; (j; ((j ( E(E ; E) ^ C)) )) )
_*__//_ * * L 0 " W
' FW [G; (j; ((j ( E(E ; E) ^ EFNG [] ^ C)) )) )
__("1)*//_ * * " W
' FW [G; (j; (E(; ; G; U)+ ^ (j (E FNG [] ^ C)) )) )
= FW [G; (j*; (E(; ; G; U)+ ^ C))W ):
Here, the maps (" ^ 1) )1, (i )1, , *, and ("1)* are similar in origin to
the analogously named maps appearing in the description of the isomorphism of
the proposition. The first of the unlabeled isomorphisms is one of those given *
*by
Lemma II.4.14 of [24]. To understand the second unlabeled isomorphism, note
that its domain and range are derived from the spectrum j*(LE(E0; E) ^ C) by
applying the sequences of change of group functors associated to the sequences *
*of
group homomorphisms
NG G __//_G
and
NG __//_WG __//_WG =W = W G;
respectively. The composites of these two sequences of homomorphisms are equal,
and so Lemma II.4.10 of [24] provides the desired natural isomorphism between t*
*he
composites of the associated sequences of functors. The last unlabeled isomorph*
*ism
is the composite of the natural isomorphism j*; j* ~= j* arising from the fact
that j = j j; and an isomorphism from Lemma II.4.14 of [24] describing the
58 13. THE PROOF OF PROPOSITION 3.10
commutativity of certain change of group and change of universe functors. It is
easy to verify that the isomorphism fl is, in fact, induced by the weak_equival*
*ence
O. __
Propositions 2.8 and 3.6 are, roughly, the special cases of Proposition 3.10
in which, respectively, = G and the indexing universe U is complete. These
two results are derived from from Proposition 3.10 by inserting the Adams and
Wirthm"uller isomorphisms in the appropriate places:
Proofs of Propositions 2.8 and 3.6. The isomorphisms in each of these
two propositions follow from the isomorphism fl of Proposition 3.10 by inserting
the appropriate Adams isomorphisms. The observation in Definition 12.1 that
AG (G; U; U0) is always just FG (G) ensures that the required Adams isomorphism
exists in the context of Proposition 2.8. The weak equivalence of Proposition 3*
*.6
follows from the weak equivalence O of Proposition 3.10 by inserting both the
appropriate Adams isomorphism and the appropriate Wirthm"uller isomorphism.
Lemma V.10.3(iii) of [24] indicates that W has finite index in G, so the usual
suspension by a representation is not required in this instance of the Wirthm"u*
*ller
isomorphism. The weak equivalence in Proposition 2.8 requires only the appropri*
*ate
Adams isomorphism since here G = e so that the change of group functor appearin*
*g __
in the weak equivalence O of Proposition 3.10 is irrelevant in this special cas*
*e. __
Our ultimate goal is to construct a wellbehaved splitting of the map
(EF0+^ C) ___//(E(F0; F) ^ C)
for appropriately split Gspectra C and suitable pairs (F0; F) of families of s*
*ubgroups
of G. Such a splitting induces a splitting of the map
0 U 0 U
j* B; EF+ ^ CG ___//j* B; E(F ; F) ^GC:
The descriptions of the maps O and fl of Proposition 3.10 given above suggest
definitions of these splitting maps for the special case in which the pair (F0;*
* F) is
adjacent. The remainder of this section is devoted to introducing the splitting
maps for this special case and showing that they do provide the desired splitti*
*ng.
We introduce the more easily understood splitting at the morphismset level bef*
*ore
introducing the spectrumlevel splitting.
Definition 13.1.Let be a normal subgroup of G, G = G=, be a subgroup
of , and F0 be a family of subgroups of G such that 2 F0. Also, let B be
a Gspectrum indexed on U , and C be a Gspectrum indexed on U which is
geometrically split at . Recall that, if L is the tangent NG representation at*
* the
identity coset eNG of G=NG , then L = 0.
(a) The map
; U 0 U
ff (F0): j* B; E(; ; G; U)+ ^ C[]WG ___//j* B; EF+ ^ CG
13. THE PROOF OF PROPOSITION 3.10 59
is the composite
; U
j* B; E(; ; G; U)+ ^ C[] WG
__i*//_ ; U
j* B; E(; ; G; U)+ ^ C WG
__*_// j; L U
* B; ( E(; ; G; U)+ ^ C) WG
__*_// j; L 0 U
* B; ( EENG <; U>+ ^ C) WG
_~=//_ L 0 U
j* B; EENG <; U>+ ^ C NG
" 1 0 U
__~=_//_ j* B; G nNG (EENG <; U>+ ^ C)G
"i* 0 U
__~=_// j* B; (G nNG EENG <; U>+ ) ^ C)G
__"*//_ U
j* B; EF0+^ C)G:
Here, i* is derived from the map i splitting C at . The maps * and "i*are
defined in Remarks II.3.14(ii) and Lemma II.4.9 of [24]. The maps * and "* are
formed from canonical maps between universal spaces. The unlabeled isomorphism
is the adjunction isomorphism for the spectrumlevel fixedpoint functor, and "*
* 1is
the inverse of the Wirthm"uller isomorphism of Theorem 11.8. Observe that ff (F*
*0)
is natural in B and in C with respect to maps which preserve the splitting at .
(b) The Gmap
fi (F0): FW [G; (j*; (E(; ; G; U)+ ^ C[]))W _)//_(EF0+^ C)
is a composite containing several natural maps relating change of universe func*
*tors.
Thus, in describing fi (F0), we again explicitly note the change of universe fu*
*nctors
included in our fixedpoint functors. The appropriate composite is then:
FW [G; (j*; (E(; ; G; U)+ ^ C[]))W )
_i*_//_ * * W
FW [G; (j; (E(; ; G; U)+ ^ (j C) )) )
_*__// F * * L W
W [G; (j; ((j ( E(; ; G; U)+ ^ C)) )) )
_*__// F * * L 0 W
W [G; (j; ((j ( EENG <; U>+ ^ C)) )) )
_~=_// * L 0 W
FW [G; ((j ( EENG <; U>+ ^ C)) ) )
_~=_// * L 0
(FNG [G; j ( EENG <; U>+ ^ C)))
_~=_// * L 0
(j FNG [G; EENG <; U>+ ^ C))
___1_// * 0
~= (j G nNG (EENG <; U>+ ^ C))
"i * 0
__~=//_(j (G nNG (EENG <; U>+ ) ^ C))
__"_// * 0
(j (EF+ ^ C)) :
60 13. THE PROOF OF PROPOSITION 3.10
Here, the maps i* , *, *, 1, "i, and " are similar in origin to the analogous*
*ly
named maps of part (a). The three unlabeled isomorphisms are, essentially, the
inverses of the three unlabeled isomorphisms appearing in the definition of O i*
*n the
proof of Proposition 3.10. However, the isomorphisms here appear in the reverse
order from those in the proof. Observe that the map fi (F0) is natural in C with
respect to maps which preserve the splitting at .
The following result, which forms the basis for the proof of our main splitt*
*ing
theorems, indicates that the maps ff (F0) and fi (F0) provide the desired split*
*tings.
Proposition 13.2. Let be a normal subgroup of a compact Lie group G,
G = G=, be a subgroup of , and (F0; F) be a adjacent pair of families of
subgroups of G associated to . Also, let B be a Gspectrum indexed on U , and C
be a Gspectrum indexed on U which is geometrically split at . Then the composi*
*te
; U
j* B; E(; ; G; U)+ ^ C[] WG
_ff_(F0)//_ 0 U
j* B; EF+ ^ CG
_*__// j 0 U
* B; E(F ; F) ^GC
_fl//_ ; U
~= j* B; E(; ; G; U)+ ^ C WG
is the isomorphism induced by the weak WG equivalence
C[] _i__//C _!__//_ C:
Moreover, the composite
FW [G; (j*; (E(; ; G; U)+ ^ C[]))W )
_fi_(F0)//_ 0
(EF+ ^ C)
__(^1)__// 0
(E(F ; F) ^ C)
__O_// * W
' FW [G; (j; (E(; ; G; U)+ ^ C)) )
is the weak Gequivalence induced by the same weak WG equivalence. Thus, the
maps
0 U 0 U
* : j* B; EF+ ^ CG___//j* B; E(F ; F) ^GC
and
( ^ 1) : (EF0+^ C) __//_(E(F0; F) ^ C)
are split epimorphisms.
Proof. Rewrite the definitions of ff (F0) and fi (F0) replacing the spaces
EE0NG <; U>+ and EF0+ by E(E0NG <; U>; ENG <; U>) and E(F0; F), respec
tively. Observe that most of the maps in these definitions become isomorphisms
when these replacements are made, and that fl and O are constructed largely by
going backwards along these isomorphisms. Thus, fl and O undo most of what
* O ff (F0) and ( ^ 1) O fi (F0) have done. Once these cancellations are remov*
*ed
from the composites fl O * O ff (F0) and O O ( ^ 1) O fi (F0), easy diagram_ch*
*ases
confirm that these maps have the asserted form. __
SECTION 14
The proofs of the main splitting theorems
This section begins with the proofs of our main splitting theorems (Theorems
2.1, 2.4, 3.2, 3.3 and 3.8) and concludes with the proof of Corollary 2.7. All *
*of the
main splitting theorems follow from a single result which is, essentially, an e*
*xtension
of Proposition 13.2 to pairs of families (F0; F) which are not adjacent. Thus,*
* the
first step in proving these splittings is introducing the maps which play the r*
*oles
analogous to those played by the maps ff (F0) and fi (F0) in Proposition 13.2.
Definition 14.1.Let be a normal subgroup of G, G = G=, and (F0; F) be
a closed pair of families of subgroups of G. Also, let B be a Gspectrum index*
*ed
on U , and C be a Gspectrum indexed on U which is geometrically split at every
subgroup of such that 2 F0 F and = embeds as a space in U.
(a) The homomorphism
M U U
ff(F0; F): j;* B; E(; ; G; U)+ ^ C[] WG ___//j* B; EF0+^ CG
()G
is the sum over the Gconjugacy classes of subgroups of such that 2 F0 F
and = embeds as a space in U of the maps
; U 0 U
ff (F0): j* B; E(; ; G; U)+ ^ C[]WG ___//j* B; EF+ ^ CG
introduced in Definition 13.1(a). Note that ff(F0; F) is natural in B and in C *
*with
respect to maps which preserve the splitting at all the specified subgroups .
(b) The Gmap
W * W 0
fi(F0; F): FW [G; (j; (E(; ; G; U)+ ^ C[])) _)_//(EF+ ^ C)
()G
is the wedge sum over the Gconjugacy classes of subgroups of such that
2 F0 F and = embeds as a space in U of the maps
fi (F0): FW [G; (j*; (E(; ; G; U)+ ^ C[]))W _)//_(EF0+^ C)
introduced in Definition 13.1(b). Note that fi(F0; F) is natural in C with resp*
*ect to
maps which preserve the splitting at all the specified subgroups .
Our generalization of Proposition 13.2 to closed pairs of families (F0; F) *
*which
are not adjacent is then:
Theorem 14.2. Let be a normal subgroup of a compact Lie group G, G =
G=, and (F0; F) be a closed pair of families of subgroups of G. Also, let B
be a Gspectrum indexed on U , and C be a Gspectrum indexed on U which is
geometrically split at every subgroup of such that 2 F0 F and = embeds
61
62 14. THE PROOFS OF THE MAIN SPLITTING THEOREMS
as a space in U. Then the composite
W * W fi(F0;F) 0
FW [G; (j; (E(; ; G; U)+ ^ C[])) )_______//_(EF+ ^ C)
()G
_(^1)__//_ 0
(E(F ; F) ^ C)
is a weak Gequivalence. Moreover, if either B is a finite GCW spectrum or the
indexing set for the direct sum in the domain of ff(F0; F) is finite, then the *
*composite
M U ff(F0;F) U
j;* B; E(; ; G; U)+ ^ C[] WG _______//_ j* B; EF0+^ CG
()G
_*__// j 0 U
* B; E(F ; F) ^GC
is an isomorphism. Thus, the maps
0 U 0 U
* : j* B; EF+ ^ CG___//j* B; E(F ; F) ^GC
and
( ^ 1) : (EF0+^ C) __//_(E(F0; F) ^ C)
are split epimorphisms.
Note that Theorems 2.1, 3.2, and 3.8 are obvious consequences of this result.
The other two main splitting theorems, Theorems 2.4 and 3.3, can be derived from
the special case of this result in which F0is the family of all subgroups and F*
* is the
empty family of subgroups. These two remaining splitting theorems are obtained
from this special case by inserting the Adams and Wirthm"uller isomorphisms in *
*the
appropriate places (as in the proofs of Propositions 2.8 and 3.6 given in the p*
*revious
section). Note that, by Lemma V.10.3(iii) of [24], the Wirthm"uller isomorphism
used in Theorem 3.3 does not require the usual suspension by a representation.
Proof of Theorem 14.2. To prove the claims about *O ff(F0; F), it suffices
to show that this composite is an isomorphism whenever B is a finite GCW spec
trum. If the indexing set for the sum in the domain of ff(F0; F) is finite, the*
*n the
apparently stronger assertion about the composite being an isomorphism for all
B follows because, in this case, both the domain and range of the composite are
cohomology theories in B. If the indexing set is not finite, then the domain of*
* the
composite may fail to satisfy the wedge axiom for B.
Our proof employs a form of induction over the families of subgroups of G
similar to that described in Proposition V.7.5(iii) of [24]. Assume first that*
* the
pair (F0; F) of the theorem is a adjacent pair associated to the subgroup of
. If = embeds as a space in U, then the indexing sets for the domains of
ff(F0; F) and fi(F0; F) consist of the single element ()G so that ff(F0; F) = f*
*f (F0)
and fi(F0; F) = fi (F0). In this case, Theorem 14.2 is, essentially, just a res*
*tatement
of Proposition 13.2, and so is proven. If, on the other hand, = does not embed
in U as a space, then the domains of ff(F0; F) and fi(F0; F) are trivial becau*
*se
they are sums indexed on the empty set. The failure of = to embed implies
that, if H 2 F0 F, then H=H cannot embed in U as an Hspace. Theorem
6.1 therefore indicates that the spectrum (E(F0; F) ^ C) is weakly Gcontracti*
*ble.
The ranges of ff(F0; F) and fi(F0; F) are then also trivial. Thus, the conclusi*
*on of
the theorem holds whenever the pair (F0; F) is adjacent.
14. THE PROOFS OF THE MAIN SPLITTING THEOREMS 63
For the inductive step in the proof, let D be the collection of Gfamilies E*
* such
that F E F0, the pair (E; F) is closed, and the conclusion of the theorem ho*
*lds
for the pair (E; F). Order D by inclusion. Clearly, D is not empty since it con*
*tains
F. We wish to show that F0is in D. Since G has only countably many conjugacy
classes_of subgroups,_any totally ordered_subset of D has a cofinal sequence {E*
*i}.
If E = [Ei, then F E F0 and the pair (E ; F) is closed._A straightforward
argument using the fact that TelEEiis_a model for_EE_indicates that the conclus*
*ion
of the theorem holds for the pair (E ; F). Thus, Eis in D, and D must have a ma*
*ximal
element E0. If E06= F0, then there are subgroups of contained in F0 E0. Pick a
minimal such subgroup of . Let E00= E0[ {H G  (H \ )G = ()G }. Then
E00is a family of subgroups of G, F E00 F0, the pair (E00; F) is closed, and
the pair (E00; E0) is adjacent. To complete the proof of the theorem, it suffi*
*ces to
show that its conclusion holds for the pair (E00; F). This contradicts the assu*
*med
maximal nature of E0and so implies that E0= F0.
The sequence
0
E(E0; F) ___//_E(E00; F) ___//_E(E00; E0)
is a cofibre sequence. It therefore provides long exact sequences of the form
0 U * 00 U
. ._.@//_j* B; E(E ; F) ^GC___//j* B; E(E ; F) ^GC
__0*//_ 00 0 U @
j* B; E(E ; E ) ^GC____//. . .
and
n 0 U * n 00 U
. ._.@//_S ^ G=H+ ; (E(E ; F) ^ C)G___//S ^ G=H+ ; (E(E ; F) ^ C)G
_0*_// n 00 0 U @
S ^ G=H+ ; (E(E ; E ) ^ C)G__//_.;. .
where n 2 Z and H G.
For any closed pair (G0; G) of Gfamilies, denote the direct sum which forms
the domain of the map ff(G0; G) by D(G0; G). Then D(E00; F) is the direct sum of
D(E0; F) and D(E00; E0). The canonical inclusion of D(E0; F) into D(E00; F) a*
*nd
the canonical projection ss of D(E00; F) onto D(E00; E0) therefore form a short*
* exact
sequence which appears as the top row in the diagram:
D(E0; F)_______________//_D(E00;_F)____ss_______//_D(E00; E0)
ff(E0;F) ff(E00;F) ff(E00;E0)
fflffl fflffl fflffl
j* B; EE0+^ CUG___*___//_j* B; EE00+^UCG j* B; EE00+^ CUG
*  0* 00*
fflffl fflffl fflffl
j* B; E(E0; F) ^UCG_*__//j* B; E(E00; F)U^GC0__//j* B; E(E00; E0)U^:C
* G
It is easy to see that the two lefthand squares in this diagram commute. To see
that the righthand rectangle commutes, it suffices to check its commutativity *
*when
restricted to each of the summands of D(E00; F). Restricted to the summand inde*
*xed
on ()G , it clearly commutes. The commutativity of the restriction to each of t*
*he
other summands follows easily from the commutativity of the two lefthand squar*
*es
64 14. THE PROOFS OF THE MAIN SPLITTING THEOREMS
and the exactness of the bottom row. Since the maps ss and 00*O ff(E00; E0) are*
* both
surjective, the map 0*must also be surjective. Since this argument applies for *
*any
finite GCW spectrum B and the bottom row of the diagram is part of a long exact
sequence, the map must be injective. The bottom row of the diagram is therefore
a short exact sequence. The left and right vertical composites are isomorphisms,
and so the center vertical composite 0*O ff(E00; F) is also an isomorphism.
A similar argument applied to the Ghomotopy groups of the domains of the
maps fi(E0; F), fi(E00; F), and fi(E00; E0) and the long exact sequence of Gho*
*motopy
groups displayed above proves that the composite ( ^ 1) O fi(E00; F) is a weak
Gequivalence. Thus, the conclusion of the theorem holds for the pair (E00;_F),*
* and
we have the desired contradiction. __
The proof below is the last of the delayed proofs from Part I, and concludes
this section.
Proof of Corollary 2.7. Let C be a Gspectrum indexed on a universe U,
and i : U __//_U0be a linear Gisometry. Assume that C is U0split. We wish to
show that the natural inclusion
ffi : iG*(CG_)_//(i*C)G
is a split monomorphism in the stable category of spectra indexed on (U0)G . Th*
*e
orem 2.4 provides weak equivalences:
W Ad(W)
CG ' EW + ^W C[]
()
and
W Ad(W)
(i*C)G ' EW + ^W i*C[];
()
in which the first wedge is indexed on the Gconjugacy classes () of subgroups
of G such that G= embeds as a Gspace in U and the second is indexed on those
classes () for which G= embeds as a Gspace in U0. Thus, to show that iG*(CG ) *
*is
a wedge summand of (i*C)G , it suffices to show that, if G= embeds as a Gspace*
* in
U, then iG*(EW + ^W Ad(W) C[]) and EW + ^W Ad(W) i*C[] are weakly
equivalent spectra. Recall that the spectrum EW + ^W Ad(W) C[] is the orbit
spectrum Z()=W of the W spectrum Z() indexed on UG which represents the
U indexed W spectrum EW + ^ Ad(W) C[]. It is easy to see that the spec
trum iG*Z() indexed on (U0)G represents the spectrum EW + ^ Ad(W) i*C[] .
The desired weak equivalence follows from this coupled with the fact that passi*
*ng
to orbits over W commutes with the functor iG*. An easy diagram chase con
firms that the resulting identification of iG*(CG ) as a wedge summand of (i*C)*
*G_is
consistent with the canonical map ffi. __
SECTION 15
The proof of the sharp Wirthm"uller
isomorphism theorem
This section contains the proof of Theorem 11.4. This result asserts that, i*
*f G is
a finite group, U is a Guniverse, and N G, then the formally defined Wirthm"u*
*ller
family W(N; G; U) of subgroups of N is equal to the more concretely defined fam*
*ily
W0(N; G; U). Recall that, in the discussion of the Wirthm"uller isomorphism at *
*the
end of Section 11, we showed that W(N; G; U) W0(N; G; U). Thus, it suffices to
show the other inclusion. This is easily reduced to showing that, if H G, n 2 *
*Z,
K 2 W0(N; G; U), and (D0; D) is an adjacent pair of families of subgroups of K,
then the Wirthm"uller map
* : [G=H+ ^ Sn; G nN (N nK E(D0; D))]UG___//
[G=H+ ^ Sn; FN [G; N nK E(D0; D))]UG
is an isomorphism. Here, and throughout the remainder of this section, we are
working entirely in the equivariant stable category, and so omit the 1U when
referring to the suspension spectrum of a space. By a change of groups isomorph*
*ism,
the map above can be identified with the map
* : [Sn; G nN (N nK E(D0; D))]UH_//[Sn; FN [G; N nK E(D0; D))]UH:
Thus, the central issue in the proof of Theorem 11.4 is the structure of the G*
*spectra
G nN (N nK E(D0; D)) and FN [G; N nK E(D0; D)) as Hspectra. This structure is
well understood only when G is finite. For this reason, Theorem 11.4 is restric*
*ted
to finite groups.
Our description of the structure of these two Gspectra considered as Hspec*
*tra
is obtained from a double coset formula applicable to the Gspectra G nN C and
FN [G; C) derived from any Nspectrum C. In this formula, we denote by [g] the
equivalence class of an element g of G in the double coset H\G=N. As one might
expect in a double coset formula, the Nspectrum C must be converted into an
appropriate (H \ gNg1)spectrum. This is accomplished using the change of
universe functor g* derived from the action map of g on U. The spectrum g*C is a
obviously a gNg1spectrum, and thus an (H \ gNg1)spectrum.
Proposition 15.1. Let H and N be subgroups of a finite group G, and let C be
an Nspectrum indexed on a Guniverse U. Then there are natural isomorphisms
W
G nN C ~= H n(H\gNg1)g*C
[g]2H\G=N
and
Y
FN [G; C) ~= F(H\gNg1)[H; g*C)
[g]2H\G=N
65
66 15. THE SHARP WIRTHM"ULLER ISOMORPHISM THEOREM
of Hspectra. Moreover, under these isomorphisms, the Wirthm"uller map
: G nN C ___//FN [G; C)
is identified with the composite Hmap
W _ W
H n(H\gNg1)g*C ____//_ F(H\gNg1)[H; g*C)
[g]2H\G=N [g]2H\G=N
_'__// Y F
(H\gNg1)[H; g*C)
[g]2H\G=N
in which the second map is the natural inclusion of the wedge into the product.
Proof. Let {gi} be a set of representatives for the cosets in G=N. Then the
Gspectrum G nN C is isomorphic, before passage to the stable category, to the
spectrum _igi*C. If g 2 G and ggiN = gjN, then the action map
G n (G nN C)___//G nN C:
of G on G nN C takes the wedge summand of the domain indexed on g and gi to
the wedge summand of the range indexed on gj via the map
gj*(g1jggi)
g*gi*C ~=gj*(g1jggi)*C __________//gj*C
derived from the action of g1jggi2 N on C.
Analogously, the Gspectrum FN [G; C)Qcan be identified, before passage to
the stable category, with the spectrum i(g1i)*C. However, (g1i)* is natural*
*lyQ
isomorphic to gi*, so that FN [G; C) can be identified with the spectrum igi**
*C.
Under this identification, the action of G on FN [G; C) is the obvious variant *
*of the
action of G on G nN C.
Under these identifications, the Wirthm"uller map
: G nN C ___//FN [G; C)
becomes the natural map
W i Y i
g*C __//_ g*C
i i
of the wedge into the product. The isomorphisms of the proposition and the de
scription of the Wirthm"uller map in the propositionQfollow from these identifi*
*cations
merely by grouping together the terms in _igi*C and igi*C which lie in_the sa*
*me
Horbit. __
This proposition and Theorems 6.1 and 11.6 suffice for our proof of the sharp
Wirthm"uller isomorphism theorem.
Proof of Theorem 11.4. The proposition allows us to identify the map
* : [G=H+ ^ Sn; G nN (N nK E(D0; D))]UG___//
[G=H+ ^ Sn; FN [G; N nK E(D0; D))]UG
15. THE SHARP WIRTHM"ULLER ISOMORPHISM THEOREM 67
with the map
M
* : [Sn; H n(H\gNg1)g*(N nK E(D0; D))]UH___//
[g]2H\G=N
M
[Sn; F(H\gNg1)[H; g*(N nK E(D0; D)))]UH:
[g]2H\G=N
Thus, it suffices to show that the composite
[Sn; H n(H\gNg1)g*(N nK E(D0; D))]UH
___*//_[Sn; F 0 U
(H\gNg1)[H; g*(N nK E(D ; D)))]H
~= [Sn; g*(N nK E(D0; D))]UH\gNg1
is an isomorphism for some set {g} of representatives for the double coset space
H\G=N. Since the pair (D0; D) of families of subgroups of K is adjacent, there *
*is
a subgroup J of K such that D0 D = (J)K . Select a pair (E0; E) of families of
subgroups of N such that E0 E = (J)N , and observe that N nK E(D0; D) and
E(E0; E) ^ N nK E(D0; D) are homotopy equivalent Nspaces.
Three cases must be considered. If (J)N (g1Hg \ N)N , then the space
E(E0; E)^N nK E(D0; D) is obviously (g1Hg \N)contractible. In this case, both
the domain and range of the above composite vanish. Thus, we may assume that
(J)N (g1Hg \ N)N .
Now consider the case in which there is no subgroup J0 2 (J)N such that
J0 g1Hg \ N and (g1Hg \ N)=J0 embeds in U as a (g1Hg \ N)space. In
this case, the target of the composite vanishes by Theorem 6.1. To see that the
domain of the composite also vanishes, select a pair (F0; F) of families of sub*
*groups
of H such that
F0 F = {Q H  9J0 g1Hg \ N; (J0)N = (J)N and (gJ0g1)H = (Q)H }:
Note that, if Q 2 F0 F, then H=Q cannot embed in U as an Hspace because
of our assumption about the nonexistence of (g1Hg \ N)embeddings. It is easy
to check that the two spectra E(F0; F) ^ H n(H\gNg1)g*(N nK E(D0; D)) and
H n(H\gNg1)g*(N nK E(D0; D)) are Hequivalent. Thus, Theorem 6.1 implies
that the domain of our composite vanishes.
The only remaining case is that in which (J)N (g1Hg \ N)N and there is a
subgroup J0 2 (J)N such that J0 g1Hg \ N and (g1Hg \ N)=J0 embeds in U
as a (g1Hg \ N)space. By choosing g appropriately in its double coset, we may
assume that J0 = J. Since J K, K 2 W0(N; G; U), and (g1Hg \ N)=J embeds
in U as a (g1Hg \ N)space, (g1Hg)=J must embed in U as a g1Hgspace.
There is a model for the Kspace E(D0; D) in which every orbit is of the form K*
*=L
for some subgroup L of J. The existence of the embedding of (g1Hg)=J in U
then implies that g*(N nK E(D0; D)), constructed using this model for E(D0; D),
is a W00(H \ gNg1; H; U)spectrum. Thus, by Theorem 11.6, our composite is_an_
isomorphism in this final case. __
SECTION 16
The proof of the Adams isomorphism theorem
for incomplete universes
Theorem 12.2 is proven here. The first step in this proof ought to be showing
that the Adams transfer o : i*((A D)=) __//_i*Dexists. However, the general
process for constructing transfers developed in [24] does not seem adequate for
constructing o when the universe is incomplete. Rather than obscuring the proof
with the gory details of an alternative construction, we begin here by summariz*
*ing,
via axioms, the properties of o needed for this proof. The theorem is then prov*
*en
under the assumption that a transfer exists satisfying those axioms. Section 17
completes this proof by supplying the desired transfer. Throughout this sectio*
*n,
we follow the variant of the notational conventions of section II.7 of [24] int*
*roduced
in Section 12. Thus, is a normal subgroup of a compact Lie group G, G = G=,
i : U0___//U is a linear Gisometry from a trivial Guniverse U0 to a Guniver*
*se
U, D is a free Gspectrum indexed on U0, and A is the adjoint representation of
G derived from .
It turns out that the Adams transfer exists for a somewhat larger class of
spectra than the class AG (; U; U0) for which it yields an isomorphism. We begin
by introducing this larger class and stating our axioms in terms of it.
Definition 16.1.If U is a Guniverse into which =e embeds as a space,
then the Gfamily FG (; U) of subgroups consists of those H G such that H\ =
{e} and H=H embeds in U as an Hspace. Note that, for any trivial G
universe U0 which embeds in U, the Adams family AG (; U; U0) is contained in
FG (; U). Also, if P is a subgroup of G such that P , then FP (; U) is just
the intersection of FG (; U) with the set of subgroups of P . If U is a comple*
*te
Guniverse, then FG (; U) = FG (). Moreover, if = G, then, for any universe
U, FG (; U) is just the family FG (G) consisting of the trivial subgroup of G.
The first two axioms impose the obvious naturality and change of group re
strictions.
(A1) The Adams transfer o is natural with respect to maps be
tween FG (; U)spectra; that is, if f : D __//_D0is a Gmap be
tween FG (; U)spectra indexed on U, then the diagram
i*((Af)=)
i*((A D)=) __________//i*((A D0)=)
o o0
fflffl i*f fflffl
i*D__________________//i*D0
commutes in the stable category of Gspectra indexed on U.
68
16. THE PROOF OF THE ADAMS ISOMORPHISM THEOREM 69
(A2) If P G and D is an FG (; U)spectrum indexed on U0,
then the Adams transfer oG : i*((A D)=) ___//i*D, considered as
a P map of P spectra indexed on U, is P homotopic to the Adams
transfer oP : i*((A D)=) ___//i*Dobtained by regarding D as an
FP (; U)spectrum.
The third axiom describes the interaction between the transfer and smash pro*
*d
ucts. A technical lemma about orbit spectra is needed for its statement.
Lemma 16.2. Let be a normal subgroup of a compact Lie group G, G be
G=, D be a Gspectrum indexed on a trivial Guniverse U0, and Y be a G
space regarded as a Gspace via the projection G __//_G. Then there is a natur*
*al
isomorphism of spectra
Y ^ (D=) ~=(Y ^ D)=:
This lemma asserts that two functors, both of which are left adjoints, are
naturally isomorphic. It is easy to see that the corresponding right adjoints *
*are
naturally isomorphic, and the lemma follows by the uniqueness of left adjoints.
Note that, if D is an FG (; U)spectrum and Y is a GCW complex, then Y ^ D
is an FG (; U)spectrum. Our third axiom relates the Adams transfers for D and
Y ^ D.
(A3) If Y is a GCW complex regarded as a Gspace via the pro
jection G __//_G and D is an FG (; U)spectrum indexed on U0,
then the diagram
Y ^ i*((A D)=) _1^o__//_Y ^ i*D
~= ~=
fflffl o fflffl
i*((A (Y ^ D))=) _____//i*(Y ^ D)
commutes in the stable category of Gspectra indexed on U.
The vertical isomorphisms in this axiom are given by Lemma 16.2 and Propo
sition II.1.4 of [24]. Observe that this axiom applied to the case Y = S1 indic*
*ates
that the Adams transfer is preserved under suspension by a trivial representati*
*on.
The final axiom is a normalization axiom indicating that o is the obvious
dimensionshifting transfer when D is just the suspension spectrum 1U0G=H+ as
sociated to some subgroup H in FG (; U). If G is finite, then this axiom is just
the assertion made in the discussion at the end of Section 12 that o is the ord*
*inary
transfer associated to the projection G=H __//_G=H. However, to state this ax
iom properly for nonfinite groups, we must first describe the structure of the *
*orbit
spectrum (A 1U0G=H+ )=. A difficulty with the action of the group H on the
vector space A comes up in this description. Let ae : H __//_H= ~=H be the
obvious projection. The group H can act on A via the inclusion H G or via
the composite of ae and the inclusion H G. These two actions are usually not
the same since H acts trivially on A with the second action; whereas A,
regarded as a representation with the first action, is just the adjoint repres*
*enta
tion of . The second action must be used for the analysis of the orbit spectrum
70 16. THE PROOF OF THE ADAMS ISOMORPHISM THEOREM
(A 1U0G=H+ )=. As a reminder of this, we denote A with this somewhat nonstan
dard Haction as ae*A. Note that A and ae*A are isomorphic as Hrepresentations
since the composite of ae and the inclusion H H is just the identity map on H.
Lemma 16.3. If H 2 FG (; U), then there is an isomorphism
*A
i*((A 1U0G=H+ )=) ~=G nH 1USae
of Gspectra indexed on U.
Proof. By Propositions I.3.6 and I.3.8 of [24], (A 1U0G=H+ )= is isomorphic
to 1U0((A G=H+ )=). Moreover, A G=H+* ~=G nH SA . There is a natural *
projection from GnH SA to GnH Sae A. Since acts trivially on ae*A, GnH Sae A
is Gisomorphic to G nH= SA considered as a Gspace via the projection G___//*
*G.
Thus, acts trivially on G nH Sae*Aand the projection*G nH SA __//_G nH Sae*A
factors through a map (G nH SA )= __//_G nH Sae A. It is easy to see that this
last map is an isomorphism of Gspaces. Combining this spacelevel isomorphism
with the isomorphism between (A 1U0G=H+ )= and 1U0((A G=H+ )=) yields
an isomorphism
*A
(A 1U0G=H+ )= ~=G nH 1U0Sae
in the category of Gspectra indexed on U0. Applying the functor i* to this iso*
*mor
phism and composing the result with two isomorphisms derived from Proposition
II.1.4 and Lemma II.4.14 of [24] gives the isomorphism whose existence_is_asser*
*ted
by the lemma. __
Now recall from the proof of Lemma II.7.6 of [24] that the Hrepresentation
derived from the tangent space of H=H at eH is just A. Since H=H embeds in
U as an Hspace, A must be contained in U as an Hrepresention. Thus, there
is a pretransfer t : S0 __//_H nH SA in the category of Hspectra indexed on
U. Our final axiom describes the Adams transfer for 1U0G=H+ in terms of this
pretransfer.
(A4) If H 2 FG (; U), then the Adams transfer
o : i*((A 1U0G=H+ )=) ___//i*1U0G=H+
for the FG (; U)spectrum 1U0G=H+ is just the composite
*A
i*((A 1U0G=H+ )=) ~= G nH 1USae
~= G nH (1USae*A^ S0)
_1nH_(1^t)//_ 1 ae*A A
G nH (U S ^ H nH S )
~= G nH H nH (1USA ^ SA )
~= G nH S0 ~=1UG=H+ ~=i*1U0G=H+
The isomorphisms in the composite map of axiom (A4) are derived from Lemma
16.3, the fact that A and ae*A are isomorphic Hrepresentations, and an assortm*
*ent
of results from chapter II of [24].
Assume now that there is an Adams transfer defined for all FG (; U)spectra
indexed on U0 and satisfying axioms (A1) through (A4). We would like to com
plete the proof of Theorem 12.2 by simply copying the proof of Theorem II.7.1
of [24]. However, that proof employs the Wirthm"uller isomorphism repeatedly.
16. THE PROOF OF THE ADAMS ISOMORPHISM THEOREM 71
This isomorphism is not necessarily available in an incomplete universe U unless
an appropriate orbit embeds in U. Thus, our proof begins with a long sequence
of inductive arguments whose purpose is to reduce the proof of the theorem to a
case in which the appropriate orbits embed. Once this reduction is accomplished,
the Wirthm"uller isomorphism can be employed to finish the proof in the same way
that the proof of Theorem II.7.1 was finished in [24].
Proof of Theorem 12.2. Assume that D is an AG (; U; U0)spectrum in
dexed on U0. We must show that, for all P G and m 2 Z, the map
m A U0 "o* m U0
G=P+ ^ S ; ( D)= G ____//G=P+ ^ S ; (i*D) G
is an isomorphism. Since the diagram
U0 "o* U0
G=P+ ^ Sm ; (A D)= G ______//G=P+ ^ Sm ; (i*D)G

~= 
fflffl 
G=P+ ^ Sm ; (A D)= U0 ~=
G 
i  
* 
fflffl o* fflffl
G=P+ ^ Sm ; i*((A D)=) UG______//_[G=P+ ^ Sm ; i*D]UG
commutes, it suffices to show that the composite along the left side and bottom*
* of
the diagram is an isomorphism. Let ae : G __//_G be the canonical projection, a*
*nd let
P = ae1(P). Using change of group isomorphisms, we can identify this composite
with the composite
m A U0 m A U0
S ; ( D)= P ~= S ; ( D)= P
_i*_// Sm ; i A U
*(( D)=) P
_o*_// [Sm ; iU
*D]P:
Here, axiom (A2) implies that we can take o to be either the P map coming from
D regarded as an AP (; U; U0)spectrum or the Gmap coming from D regarded
as an AG (; U; U0)spectrum. Therefore, an induction over the subgroups P of G
reduces our problem to showing that the composite above is an isomorphism in the
special case where P = G and P = G.
A further induction over the cells of D reduces the proof to showing that th*
*is
composite is an isomorphism when D = 1U0G=H+ , for H 2 AG (; U; U0). This
reduction uses the naturality axiom (A1). It also uses the smash product axiom
(A3) to shift the dimension of the cells of D over to the dimension of the sphe*
*re
in the domain. We prove the apparently more general result that, for any GCW
complex Y , the composite
m A 1 U0 m A 1 U0
S ; Y ^ ( U0G=H+ )= G ~= S ; Y ^ ( U0G=H+ )= G
_i*_// Sm ; Y ^ i A 1 U
*(( U0G=H+ )=) G
__(1^o)*//_ m 1 U
[S ; Y ^ i*U0G=H+ ]G
72 16. THE PROOF OF THE ADAMS ISOMORPHISM THEOREM
is an isomorphism whenever H 2 AG (; U; U0). Hereafter, we refer to this com
posite as bo(Y; H). We carry out this proof using an induction over the subgrou*
*ps
H of G in AG (; U; U0). Thus, in proving that bo(Y; H) is an isomorphism, we can
assume that bo(Y; K) is an isomorphism for every proper subgroup K of H.
For each H 2 AG (; U; U0), we show that bo(Y; H) is an isomorphism using an
induction over the families of subgroups of G. This induction reduces the probl*
*em
to showing that, for each pair (F0; F) of adjacent families of subgroups of G, *
*the map
bo(E(F0; F) ^ Y; H) is an isomorphism. Since the pair (F0; F) is adjacent, the*
*re is
a subgroup P of G such that F0 F = (P)G. If (P)G (H=)G, then both
the domain and range of the map bo(E(F0; F) ^ Y; H) vanish. To see that the
domain vanishes, note that the Gspace E(F0; F) ^ Y ^ (A G=H+ )= is weakly
contractible since ((A G=H+ )=)P is a point if (P)G (H=)G. To see that the
range vanishes, note that the Gspace E(F0; F) ^ Y ^ G=H+ is Ghomeomorphic to
G nH (E(F0; F) ^ Y ) and that E(F0; F) is Hcontractible unless (P)G (H=)G.
We may therefore assume that (P)G (H=)G and may then pick P so
that P H=. If G=P does not embed in U0, then the domain and range of
bo(E(F0; F) ^ Y; H) also vanish. The vanishing of the domain follows directly f*
*rom
Lemma 6.2. To see that the range vanishes, consider the two Gfamilies
E2 = {K G  (K)G (H)G and (K=)G (P)G}
and
E1 = {K G  (K)G (H)G and (K=)G < (P)G}:
A check of fixedpoint sets reveals that the two canonical maps
(EE2)+ ^ E(F0; F) ^ Y ^ G=H+ ___//E(F0; F) ^ Y ^ G=H+
and
(EE2)+ ^ E(F0; F) ^ Y ^ G=H+ _^1^1^1__//E(E2; E1) ^ E(F0; F) ^ Y ^ G=H+
are Gequivalences. Thus, the range of bo(E(F0; F) ^ Y; H) is isomorphic to
[Sm ; E(E2; E1) ^ E(F0; F) ^ Y ^ i*1U0G=H+U]G:
Theorem 6.1 will imply that this morphism set vanishes if we can show that, for
every K in E2 E1, the orbit G=K does not embed in U as a Gspace. A subgroup
K in E2  E1 is subconjugate to H, and it suffices to consider the case in which
K H. Since K 2 E2 E1, (K=)G = (P)G. But then, if G=P doesn't embed in
U0 as a Gspace, G=K cannot embed in U0 as a Gspace. Since H 2 AG (; U; U0)
and K H, this implies that G=K does not embed in U as a Gspace.
We now need to consider only the case in which G=P embeds as a Gspace in
U0. Using axiom (A3) and the equivariant homeomorphisms
E(F0; F) ^ Y ^ (A G=H+ )= ~= (A G=H+ ^ E(F0; F) ^ Y )=
~= (A G nH (E(F0; F) ^ Y ))=
and
E(F0; F) ^ Y ^ G=H+ ~=G nH (E(F0; F) ^ Y );
16. THE PROOF OF THE ADAMS ISOMORPHISM THEOREM 73
the map bo(E(F0; F) ^ Y; H) can be identified with the composite
m A 1 0 U0
S ; ( U0G nH (E(F ; F) ^ Y ))=G
~= Sm ; (A 1U0G nH (E(F0; F) ^ Y ))=U0G
_i*_// Sm ; i A 1 0 U
*(( U0G nH (E(F ; F) ^ Y ))=)G
_o*_// [Sm ; i 1 0 U
*U0G nH (E(F ; F) ^GY;)]
where o is the Adams transfer associated to the spectrum 1U0G nH (E(F0; F) ^ Y *
*).
Consider the two Hfamilies
E4 = {K H  (K=)G (P)G}
and
E3 = {K H  (K=)G < (P)G}:
It is easy to show that E(F0; F), considered as an Hspace, is Hequivalent to
E(E4; E3). Thus, it suffices to show that the composite
m A 1 U0
S ; ( U0G nH (E(E4; E3) ^ Y ))=G
~= Sm ; (A 1U0G nH (E(E4; E3) ^ Y ))=U0G
_i*_// Sm ; i A 1 U
*(( U0G nH (E(E4; E3) ^ Y ))=)G
_o*_// [Sm ; i 1 U
*U0G nH (E(E4; E3) ^ YG)]
is an isomorphism. This we prove by induction over the Hcells of the HCW
complex E(E4; E3)^Y . All of the Hcells of this space are constructed from sph*
*eres
of the form H=K+ ^ Sn, where K 2 E4. Thus, we have reduced the problem
to showing that the map bo(S0; K) is an isomorphism for K 2 E4. If K is a
proper subgroup of H, then this map is an isomorphism by our induction over the
subgroups of G in AG (; U; U0).
The proof has now been reduced to showing that bo(S0; H) is an isomorphism
when H 2 E4. In this case, P = H= and G=P ~= G=H embeds as a G
space in U0. Since H 2 AG (; U; U0), G=H must embed as a Gspace in U. From
this point on, the argument given in [24] for Theorem II.7.1 can be used with o*
*nly
minor adjustments to compensate for the replacement of U by U0and the different
treatment of the suspension by A.
To show that bo(S0; H) is an isomorphism, it, of course, suffices to show th*
*at
the adjoint
"o: (A 1U0G=H+ )= ___//(i*i*1U0G=H+ )
of the Adams transfer
o : i*((A 1U0G=H+ )=) ___//i*1U0G=H+
74 16. THE PROOF OF THE ADAMS ISOMORPHISM THEOREM
is a weak Gequivalence. Axiom (A4) allows us to identify o as the composite
*A
i*((A 1U0G=H+ )=) ~= G nH 1USae
~= G nH (1USae*A^ S0)
_1nH_(1^t)//_ 1 ae*A A
G nH (U S ^ H nH S )
~= G nH H nH (1USA ^ SA )
~= G nH H nH S0
~= G nH S0 ~=1UG=H+ ~=i*1U0G=H+ :
Since H=H embeds as an Hspace in U and A is the Hrepresentation derived
from the tangent space at eH of H=H, the Wirthm"uller map
: H nH S0 ___//FH [H; 1USA )
is a weak Hequivalence in the category of Hspectra indexed on U. Thus, it
suffices to show that the adjoint of the composite
*A 1 ae*A 0
G nH 1USae ~= G nH (U S ^ S )
_1nH_(1^t)//_ 1 ae*A A
G nH (U S ^ H nH S )
~= G nH H nH (1USA ^ SA )
~= G nH H nH S0
__1nH___// G n 1 A
H FH [H; U S )
is a weak Gequivalence. Just as in the proof of Theorem II.7.1 of [24], this c*
*om
posite may be identified with the map
*A 1 A
1 nH : G nH 1USae __//_G nH FH [H; U S );
where : 1USae*A___//FH [H; 1USA )is the coaction map of H on 1USae*A.
Let L be the Hrepresentation derived from the tangent space of G=H at
eH. Since G=H embeds as a Gspace in U0, it also embeds in U. Thus, the map
: G nH FH [H; 1USA ) ___//FH [G; LFH [H; 1USA ))
is a weak Gequivalence in the category of Gspectra indexed on U. It therefore
suffices to show that the adjoint of the composite
*A 1nH 1 A
G nH 1USae _______//_G nH FH [H; U S )
___//_ F L 1 A
H [G; FH [H; U S ))
~= FH [G; 1USA+L )
is an isomorphism. As in the proof of Theorem II.7.1 of [24], this adjoint is *
*the
composite
FH= [1;j) * 1 A+L
G nH= 1U0SA ___//_FH= [G; 1U0SA+L ) __________//_FH= [G; i i*U0S );
where j is the unit of the (i*; i*)adjunction. Since G=(H=) ~=G=H embeds
in U0 as a Gspace, the map is a weak Gequivalence. Also, since H= ~=H
and the universes U0 and U are Horbit equivalent by Lemma 12.3, the map j is a
weak H=equivalence by Theorem 1.2(b) of [20]. It follows that the composite_
is a weak Gequivalence, as required. __
SECTION 17
The Adams transfer for incomplete universes
Here, an ad hoc construction of the dimensionshifting Adams transfer is giv*
*en.
The notation of Section II.7 of [24], as modified in Section 12, is used throug*
*hout
this section. Recall the family FG (; U) of subgroups of G introduced in Defini*
*tion
16.1 and the axioms (A1) through (A4) for the Adams transfer presented in Secti*
*on
16. This entire section is devoted to proving the following existence theorem *
*for
the Adams transfer.
Theorem 17.1. Let be a normal subgroup of a compact Lie group G, A be
the adjoint representation of G derived from , U be a Guniverse into which =e
embeds as a space, U0 be a trivial Guniverse, and i : U0___//U be a linear G
isometry. Then, for each FG (; U)spectrum D indexed on U0, there is an Adams
transfer
o : i*((A D)=) ___//i*D:
Moreover, this transfer satisfies axioms (A1) through (A4).
Several auxiliary groups must be introduced for the proof of this theorem. In
particular, let G = G xc be the semidirect product of G and formed using the
conjugation action of G on . Let N be the subgroup e xc of G so that N is
a normal subgroup of G. Identify G with the subgroup G xc e of G. Define the
homomorphisms ffl : G __//_G and : G __//_G by
ffl(g; u) = g and (g; u) = gu:
This notation deviates from that in section II.7 of [24] in that the groups den*
*oted
here by G, N, and are denoted there by , , and N, respectively. Recall from
[24] that there is an action of G on given by
(g; u)v = guvg1 forg 2 G and u; v 2 ;
and that, with this action, is Ghomeomorphic to G=G. In [24], this orbit was
consistently referred to as (or, rather N in the notation of [24]) to compacti*
*fy
notation. Here, however, we denote it by G=G because viewing it as a coset space
clarifies the definitions of several of our maps. If X is a Gspace, Gspectru*
*m,
or some other kind of Gobject, then ffl*X and *X denote the Gobjects derived
from X via the homomorphisms ffl and . We always regard the Guniverse U as
a Guniverse via ffl. Note that N acts trivially on ffl*U so that we may form *
*N
orbit spectra indexed on ffl*U. Since acts trivially on U0, ffl*U0 and *U0 are*
* the
same Guniverse, and the inclusion i : *U0 ___//ffl*Uis a Gmap. Thus, if D is a
Gspectrum indexed on U0, then i**D is a welldefined Gspectrum indexed on
the Guniverse ffl*U. The importance of i**D comes from the isomorphisms
(i**D ^ G=G+ )=N ~=i*D and (i**D)=N ~=i*(D=);
75
76 17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES
provided by Lemma II.7.4 of [24]. These maps play a key role in the construction
of the Adams transfer.
In [24], an essential step in constructing the Adams transfer for a complete*
* G
universe U is embedding ffl*U into a complete Guniverse U00(called, unfortunat*
*ely,
U0 in [24]) via a Gisometry k : U __//_U00. Since G=G embeds in U00as a Gspac*
*e,
there is a pretransfer t : S0___//A 1U00G=G+ in the category of Gspectra inde*
*xed
on U00. If D is a free Gspectrum indexed on U0, then the induced map
1 ^ t: k*i**D ^ S0___//k*i**D ^ A 1U00G=G+
can be pulled back to a Gmap
bo: i**D ___//A (i**D ^ G=G+ )
between Gspectra indexed on U. In [24], the transfer o : i*(D=) ___//A i*D is
obtained from boby passing to orbits over N and invoking the isomorphisms recal*
*led
above from Lemma II.7.4 of [24].
If U is an incomplete Guniverse, then two difficulties arise in this approa*
*ch to
forming the Adams transfer. The most obvious is that A need not be a subrepre
sentation of U so that desuspension by A may not be defined in the category of
Gspectra (or Gspectra) indexed on U. As noted in Section 12, this difficulty *
*is
easily avoided by replacing the map boof [24] by a map of the form
bo: i**A D __//_i**D ^ G=G+
from which one derives an Adams transfer of the form
o : i*((A D)=) ___//i*D
rather than of the form
o : i*(D=) ___//A i*D
described in [24].
The more serious difficulty is that there does not seem to be a general meth*
*od
for forming a Guniverse U00that is both complete enough to admit an embedding
of G=G as a Gspace and, at the same time, incomplete enough to allow the map
1 ^ t (or, rather, A (1 ^ t) since we wish to eliminate the desuspension proble*
*m)
to be pulled back to obtain bo. In general, we resolve this difficulty by intro*
*ducing
an entirely new approach to forming the map bo. However, before introducing this
alternative method, we record here the one situation in which the method of [24]
suffices for forming the Adams transfer for an incomplete Guniverse U.
Construction 17.2. Assume that = G, so that U0 is a trivial Guniverse
and U is a Guniverse into which the free orbit G=e embeds as a Gspace. Let V *
*be
a finitedimensional subrepresentation of U such that G=e embeds as a Gspace i*
*n V
and such that V G is the onedimensional trivial Grepresentation R. Let U00be *
*the
vector space hom (*V; ffl*U) of all linear transformations from V to U, regarde*
*d as
a Gspace via the usual conjugation action on a function space. Topologize U00w*
*ith
the usual colimit topology derived from its finitedimensional subspaces. Clear*
*ly,
U00is a Guniverse. The canonical projection of V onto its fixedpoint subspace*
* R
induces a linear injection
k : ffl*U ~=hom (R; ffl*U)//_U00
which is Gequivariant, and can be made into an isometry by the choice of an
appropriate inner product on U00. Let O : V ___//U be the inclusion of V into U.
17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES 77
The Gisotropy subgroup of O, regarded as an element of U00, is G, so that G=G
embeds in U00as a Gspace. Thus, in the category of Gspectra indexed on U00,
there is a pretransfer
t : 1U00SA___//1U00G=G+:
Since is G, FG (; U) is just FG (G) and FG (; U)spectra are just Gfree
Gspectra. If D is a Gfree Gspectrum indexed on U0, then Theorem 9.3 can be
used to pull the Gmap
1 ^ t: k*i**D ^ 1U00SA___//k*i**D ^ 1U00G=G+
between Gspectra indexed on U00back to a Gmap
bo: i**A D __//_i**D ^ G=G+
between Gspectra indexed on U. The isomorphisms of Lemma II.7.4 of [24] may
then be applied just as they were in [24] to obtain the Adams transfer
o : i*((A D)=G) __//_i*D
Since this transfer is formed using the general approach to constructing transf*
*ers
presented in [24], the general results in [24] concerning the properties of tra*
*nsfers
suffice to show that o satisfies axioms (A1) through (A4).
One might hope to generalize this process to obtain an Adams transfer for any
normal subgroup of G simply by taking V to be a suitable Grepresentation
into which =e embeds as a space. Unfortunately, in this more general con
text, it seems very hard to decide which Gorbits embed in U00= hom (*V; ffl*U).
It therefore seems impossible to ensure that Theorem 9.3 can be applied to pull
back 1 ^ t to obtain bo. The key to our alternative procedure for obtaining the
map bois the observation that, if D is an FG (; U)spectrum, then the projection
D ^ EFG (; U)+ ___//Dis a Gequivalence. Thus, if we can form the Gmap
bo: i**A 1U0EFG (; U)+ ___//i**1U0EFG (; U)+ ^ G=G+
for the single FG (; U)spectrum 1U0EFG (; U)+ indexed on U0, then we can
obtain the Gmap bofor every other FG (; U)spectrum D indexed on U0 as the
composite
i**A D ' i**D ^ i**A 1U0EFG (; U)+
_1^bo//_i * *1
* D ^ i* U0EFG (; U)+ ^ G=G+
' i**D ^ G=G+ :
Once we have the Gmap bofor every FG (; U)spectrum D indexed on U0, the
Adams transfer
o : i*((A D)=) ___//i*D
can be formed as the composite
bo=N *
i*((A D)=) ~=(i**A D)=N _____//_(i* D ^ G=G+ )=N ~=i*D;
where the isomorphisms are those given by Lemma II.7.4 of [24].
Defined in this way, the Adams transfer obviously satisfies axioms (A1) and
(A3) of Section 16. Thus, to complete the proof of Theorem 17.1, we need only
construct the map bofor the spectrum 1 EFG (; U)+ and verify that the resulting
Adams transfer satisfies axioms (A2) and (A4). To see what is required to verify
78 17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES
axiom (A2), assume that P is a subgroup of G such that P , and let P be the
subgroup P xc of G. Identify P with the subgroup P xc e of P. The inclusion
of P into G induces a Phomeomorphism P=P ___//G=G. The homomorphisms
ffl : G __//_G and : G __//_G, restricted to P, provide maps into P for which *
*we
continue to use the names ffl and . The space EFG (; U), regarded as a P space,
is P equivalent to EFP (; U). Thus, to verify axiom (A2), it suffices to show *
*that,
under these identifications of G=G with P=P and of EFG (; U) with EFP (; U),
the Gmap
bo: i**A 1U0EFG (; U)+ ___//i**1U0EFG (; U)+ ^ G=G+;
regarded as a Pmap, is just the map
bo: i**A 1U0EFP (; U)+ ___//i**1U0EFP (; U)+ ^ P=P+
from which the Adams transfer for FP (; U)spectra is constructed.
The map bofor the spectrum 1U0EFG (; U)+ is constructed from a spacelevel
Gmap
*A 1 1
ffiG: *EFG (; U)+ __//_ U U G=G+ :
There is a spectrumlevel Gmap
"ffiG: i**A 1U0EFG (; U)+ ___//1UG=G+
adjoint to ffiG , and bois the composite
i**A 1U0EFG (; U)+ ___//i**1U0EFG (; U)+ ^ i**A 1U0EFG (; U)+
_1^"ffiG//_i *1 1
* U0EFG (; U)+ ^ U G=G+
~= i**1U0EFG (; U)+ ^ G=G+
in which the first map is derived from the diagonal map for the space EFG (; U).
This reduction of the construction of the Adams transfer to the construction of*
* the
Gmap ffiG also reduces the verification of axiom (A2) to showing that ffiG , r*
*egarded
as a Pmap, can be identified with the Pmap
*A 1 1
ffiP: *EFP (; U)+ __//_ U U P=P+
from which the Adams transfer for P G is constructed.
We form the Gmap ffiG by employing a simplified version of methods used in
[26] to produce approximation maps for loop spaces. The first step in defining *
*ffiG is
forming a very simple model for the space EFG (; U) out of the universe U itsel*
*f.
Definition 17.3.If V is a Grepresentation into which =e embeds as a 
space, then let V [] be the set {z 2 V  Gz \ = {e}}, where Gz is the isotropy
subgroup of the point z. Note that V [] is a Gspace. If H G, then the set
V []H is nonempty if and only if H \ = {e} and the orbit H=H embeds in V
as an Hspace. Moreover, if V []H is nonempty, then
V []H = {z 2 V  (H)z = H}:
Proposition 17.4. If is a normal subgroup of a compact Lie group G and
U is a Guniverse into which =e embeds as a space, then the Gspace U[] is
Gequivalent to FG (; U).
17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES 79
Proof. The space U[] is the colimit of the collection of spaces V [] indexed
on the finitedimensional subrepresentations V of U. For such a V , V [] clear*
*ly
has the Ghomotopy type of a GCWcomplex. Moreover, if V W , then the
inclusion of V [] into W [] is a Gcofibration. Thus, U[] has the Ghomotopy
type of a GCWcomplex. If H =2FG (; U), then U[]H = ;. If H 2 FG (; U),
then U[]H = {z 2 U  (H)z = H}, and so U[]H is contractible by the lemma
below. It follows that U[] is an FG (; U)space and that the canonical map_
U[] __//_FG (; U) is a Gequivalence. __
Lemma 17.5. Let K be a compact Lie group, J be a subgroup of K, and U
be a Kuniverse. If the space UJ = {z 2 U  Kz = J} is nonempty, then it is
contractible.
Proof. The space UJ clearly has the homotopy type of a CW complex, so it
suffices to show that, for every n 0, any map f : Sn __//_UJextends to a map
"f: Dn+1 __//_UJ. The map f, regarded as a map into U, must factor through some
finitedimensional subrepresentation V of U. Since U is a universe, there is an*
*other
subrepresentation V 0of U such that V and V 0are isomorphic and V ? V 0. Select
z 2 VJ0. Regard Dn+1 as Sn x I=Sn x {0}, and define "f: Dn+1 ___//V V 0by
"f(s; t) = (tf(s); (1  t)z), for s 2 Sn, t 2 I. Since f(s) 2 VJ for all s 2 Sn*
* and
z 2 VJ0, "f(Dn+1) (V V 0)J UJ. ___
If V is a finitedimensional subrepresentation of U, then there is a reasona*
*bly
obvious Gmap
*A ffl*Vffl*V
ffiG (V:)*V []+ ___// G=G+
which sends each z 2 V [] to a map Sffl*V +*A_//_ffl*VG=G+ derived from the
map collapsing out the complement in SV of a tubular neighborhood of the em
bedding of Gz=Gz into V that sends eGz to z. Since U[] is the colimit of
the spaces*V [], where V runs over the finitedimensional subrepresentations of
U,*and* A 1U1UG=G+*is the colimit of the corresponding collection of spaces
A fflfVflGV=G+ , we would like to form the Gmap
*A 1 1
ffiG: *EFG (; U)+ ' *U[]+ ___// U U G=G+
from the maps ffiG (V ) by passage to colimits. Unfortunately, it does not seem
possible to construct the maps ffiG (V ) in such a way that they commute with t*
*he
structure maps in these two colimit diagrams. Thus, ffiG must be constructed in*
* a
more indirect fashion.
For this indirect approach, we first form a collection of Gspaces CG (V ), *
*indexed
on the finitedimensional subrepresentations V of U, together with a collection
of inclusions of Gspaces jG (V; W ): CG (V )__//CG (W )indexed on the pairs of
subrepresentations V; W of U such that V W . Then we introduce a collection of
Gequivalences
ffG (V:)CG (V )__//V [];
and a collection of Gmaps
*A ffl*Vffl*V
fiG (V:)*CG (V )+___// G=G+
80 17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES
such that the diagrams
ffG(V )
CG (V )_____//_V []
jG(V;W) T
fflfflffG(W)
CG (W )_____//W []
and
fiG(V )* * *
*CG (V )+_____//_ A fflfVflGV=G+
*jG(V;W) _j
fflfflfiG(W)* * fflffl*
*CG (W )+ _____// A fflfWflGW=G+
commute for each pair V; W with V W . Here, __jis the usual inclusion. Let CG *
*(U)
be the colimit of the diagram formed from the spaces CG (V ) and the inclusions
jG (V; W ). The maps ffG (V ) and fiG (V ) induce a Gmap
ffG: CG (U)___//U[]
and a Gmap
*A 1 1
fiG: *CG (U)+ ___// U U G=G+ :
The map ffG is a weak Gequivalence because the maps ffG (V ) are Gequivalence*
*s,
and so the Whitehead theorem provides a Gmap flG: U[] ___//CG (U)right inverse
to ffG . Our Gmap ffiG is the composite
*flG fiG *
*EFG (; U)+ ' *U[]+ ______//*CG (U)+ ____//_ A 1U1UG=G+ :
If P is a subgroup of G such that P , then, for each finitedimensional
subGrepresentation V of U, there is a P equivalence PG(V ): CG (V )__//CP (V*
* )
such that the diagrams
fiG(V )* * *
CG (V ) ff *CG (V )+ _____// A fflfVflGV=G+
QQQQG(V ) OO
 QQ(( *P  ~ 
PG(V ) m6V6[] G(V )fflffl = 
fflfflffP(Vm)mmmm *CP (V )+ fiP(V/)/_*A ffl*Vffl*VP=P+
CP (V )
and
PG(V )
CG (V )____//_CP (V )
jG(V;W) jP(V;W)
fflfflPG(W) fflffl
CG (W )_____//CP (W )
commute. It follows immediately that the Gmap ffiG , regarded as a Pmap, can *
*be
identified with the map ffiP from which the Adams transfer for the subgroup P is
formed. Thus, axiom (A2) holds for our Adams transfer.
The space CG (V ) is a space of embeddings resembling the spaces of embeddin*
*gs
that form the little cubes and little disks operads used in loop space theory. *
*Thus,
17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES 81
the first step in constructing CG (V ) is introducing the type of embeddings wh*
*ich it
contains. If b1;Pb2; : :;:bn are nonnegative real numbers, then we call the gra*
*ph of
the equation ni=1b2ix2i= 1 in Rn an ellipsoid. Note that we allow some of the*
* bi
to be zero so that our ellipsoids may extend infinitely along some axes in Rn. *
*We
also refer to the image of this graph under any matrix B in SO(n) as an ellipso*
*id.
Define the map
j(b1; b2; : :;:bn): Rn __//_Rn
by
j(b1; b2; : :;:bn)(x1; x2; : :;:xn) = _____1______p_P(x1; x2; : :;:xn):
1 + b2ix2i
It is easy to checkPthat j(b1; b2; : :;:bn) is an embedding of Rn into the inte*
*rior of
the ellipsoid ni=1b2ix2i= 1. Moreover, j(b1; b2; : :;:bn) acts by moving each*
* point
of Rn toward the origin along the line joining that point to the origin. Note a*
*lso
that, if bi= 0, then j(b1; b2; : :;:bn) fixes all the points on the xiaxis. In*
* particular,
the embedding j(b1; b2; : :;:bn;:0)Rn+1__//_Rn+1is just j(b1; b2; : :;:bn)x1. T*
*his
property of these embeddings makes possible the passage to colimits that is an
essential part of the definition of ffiG .
If B 2 SO(n), then let
j(b1; b2; : :;:bn;:B)Rn__//_Rn
be the composite Bj(b1; b2; : :;:bn)B1. This mapPembeds Rn into the interior of
the ellipsoid obtained by rotating the graph of ni=1b2ix2i= 1 by B. Note that
j(b1; b2; : :;:bn; B) also acts by moving each point of Rn toward the origin al*
*ong
the line joining that point to the origin. Hereafter, we refer to embeddings of
the form j(b1; b2; : :;:bn; B) as ellipsoidal embeddings. Let I(Rn) be the set *
*of all
ellipsoidal embeddings of Rn into Rn. Assigning the embedding j(b1; b2; : :;:bn*
*; B)
to the element (b1; b2; : :;:bn; B) of [0; 1)n x SO(n) gives a surjective map f*
*rom
[0; 1)n x SO(n) to I(Rn). We give I(Rn) the quotient topology derived from this
map. It is fairly easy to see that this agrees with the subspace topology that *
*I(Rn)
inherits from the space of all continuous functions from Rn to Rn. If V is a fi*
*nite
dimensional inner product space of dimension n, then we identify V with Rn by
choosing an orthonormal basis. This identification allows us to define both an
ellipsoidal embedding of V into V and the space I(V ) of all such embeddings.
The ellipsoidal embeddings that are, essentially, the points of the space CG*
* (V )
have as their domains the fibres of a vector bundle over V []. If z 2 V [], then
there is an embedding of the Gzspace Gz=Gz into V which sends the identity
coset eGz to z. LetSEz(V ) be the vector subspace of V orthogonal to this embed*
*ding
at z. Let E(V ) = z2V []Ez(V ) and define (V ) : E(V )__//_V []to be the map
sending the entire vector space Ez(V ) to z. It is easy to see that (V ) is a *
*G
vector bundle over V []. For each z in V [], the inclusion of into Gz=Gz
is a diffeomorphism. Thus, the bundle (V ) depends on and V , but not on the
ambient group G containing . This observation is critical for the proof that ax*
*iom
(A2) is satisfied. The diffeomorphism between and Gz=Gz also implies that
the tangent space of Gz=Gz at eGz is isomorphic, as a Gzrepresentation, to A.
Let OE(V ): A x E(V )__//_V x V []be the map which sends a pair (a; v) consisti*
*ng
of a 2 A and v 2 Ez(V ), for z 2 V [], to the pair (v + dza; z), where dz is the
differential of the embedding of Gz=Gz into V sending eGz to z. Clearly, OE(V )*
* is
82 17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES
a Ghomeomorphism which identifies the bundle A (V ) with the trivial bundle
V x V [] __//_V []. Denote the restriction of OE(V ) to the fibre over z 2 V []*
* by
OEz(V:)A Ez(V )___//V . Note that OEz(V ) is a Gzequivariant linear isomorphi*
*sm.
We can now describe the space CG (V ) and the map ffG (V ) precisely.
Definition 17.6.(a) Let V be a Grepresentation into which the free orbit
=e embeds as a space. The space CG (V ) is the subset of the set (A x E(V ))V
of continuous maps from V to A x E(V ) consisting of those maps satisfying the
three conditions:
(i)There is an z 2 V [] and an ellipsoidal embedding j 2 I(Ez(V )) such that
the diagram
OEz(V )
A Ez(V ) _____//Vu
uu
1j  uuuuu
fflfflzzuu
A Ez(V )
commutes.
(ii)The map is Gzequivariant, where z is the point of V [] specified in con
dition (i).
(iii)The subset z + j(Ez(V )) of V is a Gzslice at z; that is, the Gzmap
from Ez(V ) to V which sends v 2 Ez(V ) to z + j(v) extends to a Gz
homeomorphism from Gz xGz Ez(V ) to an open neighborhood of Gzz.
Of course, the map ffG (V:)CG (V )__//V []sends each 2 CG (V ) to the z 2 V []
associated with in condition (i). Observe that the set CG (V ) is a Ginvaria*
*nt
subset of (AxE(V ))V and that ffG (V ) is a Gmap. If 2 CG (V ) and z = ffG (V*
* )(),
then can be factored as the composite
0 OE1z(V )
V _j__//V ________//A Ez(V )
for some ellipsoidal embedding j0 of V into V . The assignment of the pair (z; *
*j0)
to the map 2 CG (V ) embeds the set CG (V ) into V [] x I(V ). Give CG (V )
the subspace topology derived from this embedding; this actually agrees with the
subspace topology that CG (V ) inherits from (A x E(V ))V . The map ffG (V ) is
continuous since it is just the restriction of the projection of V [] x I(V ) o*
*nto
V [].
(b) In order to show that the map ffG (V ) is a Gequivalence, we must constr*
*uct
its homotopy inverse sG (V ): V []___//CG (V.)For each z 2 V [], there is a real
number rz > 0 such that the disk of radius rz in Ez(V ) provides a slice at z. *
*There
is an ellipsoidal embedding jz(rz) : Ez(V ) __//_Ez(V ) which maps Ez(V ) onto *
*the
interior of this disk. The element sG (V )(z) of CG (V ) should be the composite
OE1z(V ) 1jz(rz)
V ________//A Ez(V ) ________//_A Ez(V ):
The only difficulty with this definition of sG (V ) is that the radii rz must b*
*e selected
so that the assignment of rz to z gives a continuous Gmap from V [] to the spa*
*ce
(0; 1), which carries a trivial Gaction. It is easy to see that a radius that *
*works
at z 2 V [] also works at each point in the Gorbit of z. Moreover, at each
point z, a radius can be selected which works for every point in a neighborhood
of z. A partition of unity argument can then be used to construct an appropriate
equivariant continuous function r : V [] __//_(0; 1).
17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES 83
Since our ellipsoidal embeddings simply move points inward along lines throu*
*gh
the origin, it is easy to see that, for each z 2 V [], every in ffG (V )1(z) *
*can be
deformed continuously to the standard embedding sG (V )(z) in ffG (V )1(z). Th*
*is
deformation can be performed continuously in z. We have thus proven the followi*
*ng
result:
Lemma 17.7. Let V be a Grepresentation into which the free orbit =e embeds
as a space. Then the maps
sG (V ): V []___//CG (V )andffG (V:)CG (V )__//V []
display V [] as a Gequivariant strong deformation retract of CG (V ).
The maps
PG(V ) : CG (V_)//_CP (V );
jG (V; W ) : CG (V_)//CG (W );
and
*A ffl*Vffl*V
fiG (V ) : *CG (V_)+//_ G=G+
must still be defined, and it must be shown that the various diagrams relating *
*these
maps to each other and to ffG (V ) commute. Also, we must prove that axiom (A4)
is satisfied.
Definition 17.8.(a) Assume that P is a subgroup of G such that P ,
and let V be a Grepresentation into which the free orbit =e embeds as a space.
The spaces CG (V ) and CP (V ) are both subsets of (A x E(V ))V . It is easy to
see that CP (V ) contains CG (V ), but is generally strictly larger. Take the P*
* map
PG(V ): CG (V )__//CP (Vt)o be just the inclusion of CG (V ) as a subset. Clear*
*ly,
ffG (V ) = ffP (V )OPG(V ). It follows from Lemma 17.7 that PG(V ) is a P equi*
*valence.
(b) If W is a Grepresentation which contains V and W  V is the orthogonal
complement of V in W , then Ez(W ) = Ez(V ) (W  V ) for each z 2 V []. Thus,
the restriction of (W ) to V [] W [] is just (V ) (W  V ). Under these
identifications, the map OEz(W ): A Ez(W ) ___//W becomes the map
OEz(V ) 1: A Ez(V ) (W  V )___//V (W  V )
whenever z 2 V []. If 2 CG (V ), ffG (V )() = z, and j : Ez(V )__//_Ez(Vi)s the
ellipsoidal embedding associated to by condition (i) of Definition 17.6(a), th*
*en
the composite
"j: Ez(W ) = Ez(V ) (W  V ) _j1__//Ez(V ) (W  V ) = Ez(W )
is also an ellipsoidal embedding. Moreover, since j(Ez(V )) provides a Gzslice*
* at
z 2 V , "j(Ez(W )) provides a Gzslice at z considered as an element of W . Thu*
*s,
the map
1(W) 1"j
": W _OEz____//A Ez(W ) _____//A Ez(W )
is an element of CG (W ). Let jG (V; W ): CG (V )__//CG (Wb)e the map given by
jG (V; W )() = ". Clearly, jG (V; W ) is a Ginclusion, ffG (V ) = ffG (W ) O j*
*G (V; W ),
and jP (V; W ) O PG(V ) = PG(W ) O jG (V; W ).
84 17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES
(c) Let be an element of CG (V ), ffG (V )() = z, and j : Ez(V )__//_Ez(Vb)e
the ellipsoidal embedding associated to by condition (i) of Definition 17.6(a)*
*. To
describe the map
*A ffl*Vffl*V
fiG (V )() 2 G=G+ ;
we must first introduce the collection of auxiliary maps from which fiG (V )() *
*is
formed. The sphere SV contains an open neighborhood of the orbit Gzz which is
Gzhomeomorphic to Gz xGz j(Ez(V )). Collapsing out the complement of this
neighborhood gives a Gzmap
p() : SV __//_Gz nGz Sj(Ez(V;))
where Sj(Ez(V ))denotes the onepoint compactification of j(Ez(V )). Let Gz =
1(Gz). The homomorphism ffl induces an isomorphism between Gz and Gz.
This isomorphism maps Gz xce Gz onto Gz Gz. Hereafter, we identify Gz
and Gzxce. From these observations about ffl and Gz, we obtain an Gzisomorphism
i j
bffl: Gz nGz Sj(Ez(V ))//_ffl* Gz nGz Sj(Ez(V:))
By regarding A as a Grepresentation via , we obtain an isomorphism
i j
i : Gz nGz SA ^ Sj(Ez(V ))_//_SA ^ Gz nGz Sj(Ez(V ))
which is the spacelevel analog of the isomorphism i of Lemma II.4.9 of [24]. A*
*lso,
let
i0: Gz nGz SV __//_(Gz=Gz)+ ^ SV
be the analogous isomorphism obtained by regarding V as a Gspace via ffl. The
map : V ___//A Ez(V )induces a Gzhomeomorphism
b : SV __//_SA ^ Sjz(E(V:))
The inclusion of Gz into G induces a homeomorphism from Gz/Gz to G=G which
we denote by .
The composite
1^p() A j(E (V ))
SA ^ SV _______//S ^ Gz nGz S z
_1^bffl1//_SA ^ G j(Ez(V ))
z nGz S
_i1_// G i A j(Ez(V ))j
z nGz S ^ S
_1nb1_//G V
z nGz S
__i0//_ (G V
z=Gz)+ ^ S
_^1__// G=G V
+ ^ S
may be regarded as an element of *A ffl*Vffl*VG=G+ , and we define fiG (V ) by
letting fiG (V )() be this element. The continuity of fiG (V ) is most easily c*
*hecked
by looking at its adjoint
f"iG(V:)*CG (V )+ ^ S*A ^ Sffl*V//_G=G+ ^ Sffl*V:
A somewhat messy diagram chase indicates that fiG (V ) is a Gmap. In chasing t*
*his
diagram, it is essential to remember that the action of G on *CG (V )+ is deriv*
*ed
17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES 85
from the action of G on V via , whereas the action of G on *A ffl*Vffl*VG=G+
is derived from the action of G on V via ffl.
To verify that the diagram relating fiG (V ), fiP (V ), and PG(V ) commutes,*
* note
that replacing Gz by Pz in the composite specifying fiG (V )() does not alter t*
*he
composite at all because, at every stage of the composite in which Gz or Gz
appears, the inclusion Pz Gz induces isomorphisms between the spaces ap
pearing in the diagram and the spaces obtained by replacing Gz by Pz or Gz
by 1(Pz). This completes the proof that our transfer satisfies axiom (A2).
To see that __jO fiG (V ) = fiG (W ) O *jG (V; W ), consider the adjoints of*
* these
two maps. Observe that both of these adjoints are just the composite
*A ffl*W * *A ffl*V ffl*(WV )
*CG (V )+ ^ S ^ S ~= CG (V )+ ^ S ^ S ^ S
_"fiG(V_)^1//_ * *
G=G+ ^ Sffl V^ Sffl (WV )
~= G=G+ ^ Sffl*W;
where "fiG(V ) is the adjoint of fiG (V ) that was used to check the continuity*
* of fiG (V ).
The following result completes the last step in the proof of Theorem 17.1.
Lemma 17.9. The transfer derived from the map ffiG satisfies axiom (A4).
Proof. Let H 2 FG (; U). We must show that the Adams transfer
o : i*((A 1U0G=H+ )=) ___//i*1U0G=H+
for the FG (; U)spectrum 1U0G=H+ is the composite specified by the axiom. The
projection
G=H+ ^ U[]+ ' G=H+ ^ FG (; U)+ ___//G=H+
from which the transfer for 1U0G=H+ is derived has an obvious homotopy in
verse derived from the Gmap f : G=H __//_U[]whose existence is guaranteed
by the universal properties of U[]. Since CG (U)H is connected and contains
points with isotropy subgroup H, we can select f so that the composite flG O f:
G=H __//_CG (U)takes the identity coset eH to an element of CG (U) with isotr*
*opy
subgroup H. Let z = ffG (). Note that z also has isotropy subgroup H. Since
G=H is compact, there is a finitedimensional subrepresentation V of U such that
f(G=H)) V [] and flG (f(G=H)) CG (V ). Then is in CG (V ), and z is in
V [].
Let
bffi: *G=H+ ^ S*A ^ Sffl*V__//G=G+ ^ Sffl*V
be the adjoint of the composite
fiG(V ) *A ffl*Vffl*V
*G=H+ ___//*CG (V )+ _______// G=G+
in which the first map is the unique Gmap sending eH to . The Gmap
bo: i**A 1U0G=H+ __//_i**1U0G=H+ ^ G=G+
from which we construct the Adams transfer is just the stabilization of the com
posite
*A ffl*V ^1^1 * * *A ffl*V
*G=H+ ^ S ^ S _______//_ G=H+ ^ G=H+ ^ S ^ S
_1^bffi//_*G=H ffl*V
+ ^ G=G+ ^ S ;
86 17. THE ADAMS TRANSFER FOR INCOMPLETE UNIVERSES
which we denote by __o. There are isomorphisms
*V V
(*G=H+ ^ G=G+ ^ Sffl)=N ~=G=H+ ^ S
and
*A ffl*V A V
(*G=H+ ^ S ^ S )=N ~=(( G=H+ )=) ^ S
which are the spacelevel analogs of the isomorphisms of Lemma II.7.4 of [24].
Composing these isomorphisms with the map obtained from __oby passage to N
orbits, we obtain the map
o1: ((A G=H+ )=) ^ SV ___//G=H+ ^ SV
whose stabilization is the Adams transfer
o : i*((A 1U0G=H+ )=) ___//i*1U0G=H+:
The map which axiom (A4) asserts must be equal to the Adams transfer for
1U0G=H+ is easily seen to be the stabilization of the composite
*A V
((A G=H+ )=) ^ SV ~= (G nH Sae ) ^ S
~= G nH (Sae*A^ SV )
_1n(1^p())//_ ae*A j(E (V ))
G nH (S ^ (H nH S z ))
~= G nH H nH (SA ^ Sj(Ez(V )))
_1n1nb1_// G n V
H H nH S
~= G=H+ ^ SV ;
which we denote o2. Here, the first isomorphism is one described in the proof of
Lemma 16.3. Also, as defined just above that lemma, ae*A is just A with the H
action derived from the projection ae : H __//_H= ~=H rather than the usual
action derived from the inclusion of H into G. The other unlabeled isomorphisms
in this composite are spacelevel versions of the isomorphism i of Lemma II.4.9*
* of
[24]. The remaining maps in the composite are all described in Definition 17.8(*
*c).
Since the Adams transfer for 1U0G=H+ and the map to which it is compared
in axiom (A4) are the stabilizations of o1 and o2, axiom (A4) can be verified by
showing that o1 and o2 are the same Gmap. This follows from a rather tedious
diagram chase which*can be simplified by identifying the domain of both maps
with G nH (Sae A^ SV ). The universal property of this space reduces the proof*
* to
establishing the equality of the*two maps derived from*o1 and o2 by precomposit*
*ion_
with the canonical inclusion Sae A^ SV ___//G nH (Sae A^ SV ). _*
*_
Acknowledgments
My work on these splitting results was inspired by a sequence of conversatio*
*ns
with Tammo tom Dieck about his own work, and the work of his student Henning
Hauschild, on various precursors of Theorems 2.4, 3.3, and 3.8. I would like to*
* ex
press my gratitude to tom Dieck for the insights he offered in those conversati*
*ons,
and to the Alexander von Humboldt Foundation and Sonderforschungsbereich 170
for the support and hospitality that made those conversations possible. I am de*
*eply
indebted to Peter May for many helpful comments on several drafts of this mem
oir. The hospitality of Haynes Miller, Mike Hopkins, and the MIT Mathematics
Department during the research leave on which I completed the manuscript is also
greatly appreciated.
87
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