EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA
H. FAUSK
Abstract.We extend the theory of equivariant orthogonal spectra from fi
nite groups to profinite groups, and more generally from compact Lie gro*
*ups to
compact Hausdorff groups. The Ghomotopy theory is "pieced together" from
the G=Uhomotopy theories for suitable quotient groups G=U of G; a motiv*
*a
tion is the way continuous group cohomology of a profinite group is buil*
*t out of
the cohomology of its finite quotient groups. In this category Postnikov*
* tow
ers are studied from a general perspective. We introduce proGspectra a*
*nd
construct various model structures on them. A key property of the model *
*struc
tures is that prospectra are weakly equivalent to their Postnikov tower*
*s. We
give a careful discussion of two version of a model structure with "unde*
*rlying
weak equivalences". One of the versions only make sense for prospectra.*
* In
the end we use the theory to study homotopy fixed points of proGspectr*
*a.
Contents
1. Introduction 2
1.1. Acknowledgements 5
2. Unstable equivariant theory 5
2.1. GSpaces 6
2.2. Collections of subgroups of G 6
2.3. Model structures on the category of Gspaces 8
2.4. Some change of groups results for spaces 10
3. Orthogonal GSpectra 11
3.1. JGVspaces 11
3.2. Orthogonal Rmodules 13
3.3. Fixed point and orbit spectra 13
3.4. Examples of orthogonal GSpectra 14
3.5. The levelwise Wmodel structures on orthogonal GSpectra 14
4. The stable Wmodel structure on orthogonal Gspectra 15
4.1. Verifying the model structure axioms 15
4.2. Fibrations 18
4.3. Positive model structures 20
4.4. Homotopy classes of maps between suspension spectra 20
4.5. The Segaltom Dieck splitting theorem 23
4.6. Selfmaps of the unit object 23
5. The W  Cmodel structure on orthogonal Gspectra 24
5.1. The construction of WCMR 25
____________
Date: August 28, 2006.
1991 Mathematics Subject Classification. Primary 55P91; Secondary 18G55.
Key words and phrases. Equivariant homotopy, prospectra, profinite groups.
Support by the Institut MittagLeffler (Djursholm, Sweden) is gratefully ack*
*nowledged.
1
2 H. FAUSK
5.2. Tensor structures on MR 26
5.3. The Ccofree model structure on MR 28
6. A digression: Gspectra for noncompact groups 28
7. Postnikov tmodel structures 32
7.1. Preliminaries on tmodel categories 32
7.2. The dPostnikov tmodel structure on MR 33
7.3. An example: Greenlees connective Ktheory 34
7.4. Postnikov sections 34
7.5. Coefficient systems 36
7.6. Continuous Gmodules 38
8. ProGspectra 38
8.1. Examples of proGSpectra 39
8.2. The Postnikov model structure on proMR 39
8.3. Tensor structures on pro MR 42
8.4. Bredon cohomology 43
8.5. Group cohomology 44
8.6. Homotopy orbits and group homology 44
8.7. The AtiyahHirzebruch spectral sequence 45
9. The Cfree model structure on pro MR 46
9.1. Construction of the Cfree model structure on pro MR 46
9.2. Comparison of the free and the cofree model structures 47
10. Homotopy fixed points 49
10.1. The homotopy fixed points of a prospectrum 49
10.2. Homotopy orbit and homotopy fixed point spectral sequences 51
10.3. Comparison to Davis' homotopy fixed points 53
Appendix A. Compact Hausdorff Groups 55
References 56
1.Introduction
This paper is devoted to explore some aspects of equivariant homotopy theory
of Gequivariant orthogonal spectra when G is a profinite group. We develop the
theory sufficiently to be able to construct homotopy fixed points of Gspectra *
*in
a natural way. A satisfactory theory of Gspectra, when G is a profinite group,
requires the generality of proGspectra. The results needed about model struc
tures on procategories are presented in two papers joint with Daniel Isaksen [*
*19]
[20]. Most of the theory also works for compact Hausdorff groups and discrete
groups.
We start out by considering model structures on Gspaces. This is needed as a
starting point for the model structure on Gspectra. A set of closed subgroups *
*of
G is said to be a collection if it is closed under conjugation. To any collecti*
*on C of
subgroups of G, we construct a model structure on the category of Gspaces such
that a Gmap f is a weak equivalence if and only if fH , for H 2 C, is a underl*
*ying
weak equivalence.
The collections of subgroups of G that play the most important role in this
paper are the cofamilies, i.e. collections of subgroups that are closed under p*
*assing
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 3
to larger subgroups. The example to keep in mind is the cofamily of open subgro*
*ups
in a profinite group.
We present the foundation for the theory of orthogonal Gspectra, indexed on
finite orthogonal Grepresentations, with minimal assumptions on the group G
and the collection C. Most of the results extend easily from the theory develop*
*ed
for compact Lie groups by Mandell and May [35]. We include enough details to
make our presentation readable, and provide new proofs when the generalizations
to our context are not immediate. Equivariant Ktheory and stable equivariant
cobordism theory both extend from compact Lie groups to general compact Haus
dorff groups. A generalization of the AtiyahSegal completion theorem is studied
in [18].
Let R be a symmetric monoid in the category of orthogonal Gspectra indexed
on a universe of Grepresentations. In Theorem 4.7 the category of Rmodules,
denoted MR , is given a stable model structure, such that the weak equivalences*
* are
maps whose Hfixed points are stable equivalences for all H in a suitable colle*
*ction
W. For example W might be the smallest cofamily containing all normal subgroups
H of G such that G=H is a compact Lie group. A stable Gequivariant theory of
spectra, for a profinite group G, is also given by Gunnar Carlsson in [5].
We would like to have a notion of "underlying equivalence" even when the triv*
*ial
subgroup, {1}, is not included in the collection C. We consider a more general
framework. In Theorem 5.5 we show that for two reasonable collections, W and C,
of subgroups of G such that W U is in C, whenever W 2 W and U 2 C, there is a
model structures on MR such that the cofibrations are relative Ccell complexes
and the weak equivalences are maps f such that W*(f) = colimU2CssWU*(f) is
an isomorphism for every W 2 W. For example, C can be the collection of open
subgroups of a profinite group G and W the collection, {1}, consisting of the t*
*rivial
subgroup in G.
In the rest of this introduction we assume that Un(R) = 0 whenever n < 0
and U 2 W. We can then set up a good theory of Postnikov sections in MR .
The Postnikov sections are used in our construction of the model structures on
pro MR . Although we are mostly interested in the usual Postnikov sections that
cut off the homotopy groups at the same degree for all subgroups W 2 W, we give
a general construction that allow the cutoff to take place at different degrees*
* for
different subgroups.
In Theorem 8.4 we construct a stable model structure, called the Postnikov
W  Cmodel structure, on pro MR . It can be thought of as the localization
of the strict model structure on pro MR , where we invert all maps from a pro
spectrum to its levelwise Postnikov tower, regarded as a prospectrum. Here is *
*one
characterization of the weak equivalences: The class of weak equivalences in the
Postnikov W Cmodel structure is the class of promaps that are isomorphic to a
levelwise map {fs}s2S such that fs becomes arbitrarily highly connected (unifor*
*mly
with respect to the collection W) as s increases [19, 3.2].
In Theorem 8.27 we give an AtiyahHirzebruch spectral sequence. It is con
structed using the Postnikov filtration of the target prospectrum. The spectr*
*al
sequence has good convergence properties because any prospectrum can be recov
ered from its Postnikov tower in our model structure.
4 H. FAUSK
The category proMR inherits a tensor product from MR . This tensor structure
is not closed, and it does not give a welldefined tensor product on the whole
homotopy category of pro MR with the Postnikov W  Cmodel structure.
The Postnikov W  Cmodel structure on pro MR is a stable model struc
ture. But the associated homotopy category is not an axiomatic stable homotopy
category in the sense of HoveyPalmieriStrickland [27].
We discuss two model structures on pro  MR with two different notions of
"underlying weak equivalences". Let G be a finite group and let C be the collec*
*tion
of all subgroups of G. There are many different, but Quillen equivalent, W 
Amodel structures on MR with W = {1} and A C. Two extreme model
structures are the cofree model structure, with A = C, and the free model struc*
*ture,
with A = W = {1}. The cofibrant objects in the free model structure are retracts
of relative Gfree cell spectra.
Now let G be a profinite group and let C be the collection of all open subgro*
*ups
of G. In this case the situation is more complicated. The {1}weak equivalences
are maps f such that {1}*(f) = colimU2CssU*(f) is an isomorphism. We call these
maps the Cunderlying weak equivalences. Let G be a nonfinite profinite group,
and let C be the collection of all open subgroups of G. The Postnikov {1}Cmod*
*el
structure on pro MR is the closest we can get to a cofree model structure. It
is given in Theorem 8.5. Certainly, it not sensible to have a model structure w*
*ith
cofibrant objects relative free Gcell complexes, because Sn ^G+ is equivalent *
*to a
point. In pro MR , unlike MR , we can form an arbitrarily good approximation to
the free model structure by letting the cofibrations be retracts of levelwise r*
*elative
Gcell complexes that become "eventually free". That is, as we move up the inve*
*rse
system of spectra, the stabilizer subgroups of the relative cells become smalle*
*r and
smaller subgroups in the collection C. The key idea is that the cofibrant repla*
*cement
of the constant prospectrum 1 S0 should be the prospectrum
{ 1 EG=N+ },
indexed by the normal subgroups N of G in C, ordered by inclusion. We use the
rather technical theory of filtered model categories, developed in [19], to con*
*struct
the free model structure on pro MR . This Cfree model structure is given in
Theorem 9.2.
The Cfree and Ccofree model structures on pro MR are Quillen adjoint,
via the identity maps, but there are fewer weak equivalences in the free than i*
*n the
cofree model structure. Thus, we actually get two different homotopy categories.
We relate this to the failure of having an inner hom functor in the procategor*
*y.
Let Ho(pro MR ) denote the homotopy category of pro MR with the Postnikov
Cmodel structure. Assume that X is cofibrant and that Y is fibrant in the Post
nikov Cmodel structure on pro MR . Then Theorem 9.10 says that the homset
of maps from X to Y in the homotopy category of the Cfree model structure on
pro MR is:
Ho(pro MR ) (X ^ {EG=N+ }, Y )
while the homset in the homotopy category of the Ccofree model structure on
pro MR is:
Ho (pro MR ) (X, hocolimUF (EG=N+ , Y ),
where the colimit is taken levelwise.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 5
The Postnikov model structures are wellsuited for studying homotopy fixed
points. For definiteness, let G be a profinite group, let C be the collection o*
*f open
subgroups of G, and let R be a nonequivariant Scell spectrum with trivial hom*
*o
topy groups in negative degrees. The homotopy fixed points of a proGspectrum
{Yt} is defined to be the Gfixed points of a fibrant replacement in the Postni*
*kov
Ccofree model structure. It is equivalent, in the Postnikov model structure on
Rspectra, to the prospectrum
hocolimNF (EG=N+ , PnYt)G
indexed on n and t. The spectrum associated to the homotopy fixed point pro
spectrum (take homotopy limits) turns out to be equivalent to
holimt,mhocolimNF ((EG=N)(m)+, Yt)G .
These expressions resemble the usual formula for homotopy fixed points.
The appropriate notion of a ring spectrum in proMR is a monoid in proMR .
This is more flexible than a promonoid. The second formula for homotopy fixed
point spectra shows that if Y is a (commutative) fibrant monoid in pro MR with
the strict Cmodel structure, then the associated homotopy fixed point spectrum
is a (commutative) monoid in MR .
Under reasonable assumptions there is an iterated homotopy fixed point formul*
*a.
This appears to be false if one defines homotopy fixed points in the Cstrict m*
*odel
structure on pro MR . We obtain a homotopy fixed point spectral sequence as a
special case of the AtiyahHirzebruch spectral sequence.
The explicit formulas for the homotopy fixed points, the good convergence pro*
*p
erties of the homotopy fixed point spectral sequence, and the iterated homotopy
fixed point formula are all reasons for why it is convenient to work in the Pos*
*tnikov
Cmodel structure.
A general theory of homotopy fixed point spectra for actions by profinite gro*
*ups
was first studied by Daniel Davis in his Ph.D. thesis [8]. His theory was inspi*
*red
by a homotopy fixed point spectral sequence for En, with an action by the ex
tended Morava stabilizer group, constructed by Ethan Devinatz and Michael Hop
kins [12]. We show that our definition of homotopy fixed point spectra agrees w*
*ith
Davis' when G has finite virtual cohomological dimension. Our theory applies to
the example of En above, provided we follow Davis and use the "prospectrum
K(n)localization" of En rather than (the K(n)local spectrum) En itself.
1.1. Acknowledgements. The model theoretical foundations for this paper is
joint work with Daniel Isaksen. I am also grateful to him for many discussions
on the foundation of this paper. I would like to thank Andrew Blumberg and
Daniel Davis for their interest in the paper and their help.
2.Unstable equivariant theory
We associate to a collection, W, of closed subgroups of G a model structure on
the category of based Gspaces. The weak equivalences in this model structure a*
*re
maps f such that the Hfixed points map fH is a nonequivariant weak equivalence
for each H 2 W.
6 H. FAUSK
2.1. GSpaces. We work in the category of compactly generated weak Hausdorff
spaces. Let G be a topological group. A Gspace X is a topological space togeth*
*er
with a continuous left action by G. The stabilizer of x 2 X is {g 2 G  gx = x}*
*. This
is a closed subgroup of G since it is the preimage of the diagonal in X xX unde*
*r the
map g 7! x x gx. Let Z be any subset of X. The stabilizer of Z is the intersect*
*ion
of the stabilizers of the points in Z, hence a closed subgroup of G. Similarly,*
* for any
subgroup H of G the Hfixed points, XH = {x 2 X  hx = x for eachh 2 H},
of a Gspace X is a closed subset_of X. The stabilizer of XH contains_H and is a
closed subgroup of G. So XH = XH , for any subgroup H of G, where H denotes
the closure of H in G. Hence, we consider closed subgroups of G only.
A based Gspace is a Gspace together with a Gfixed basepoint. We denote
the category of based Gspaces and basepoint preserving continuous Gmaps by
GT .
Lemma 2.1. The category of based Gspaces GT is complete and cocomplete.
Proof.The limits and colimits are created via the forgetful functor to spaces [*
*34].
We denote the category of based Gspaces and continuous basepoint preserving
maps by TG . The space of continuous maps is given a Gaction by (g . f)(x) =
gf(g1x) (and topologized as the Kellyfication of the compact open topology). T*
*he
corresponding categories of unbased Gspaces are denoted GU and UG .
Lemma 2.2. Let X and Y be two Gspaces. The action of G on TG (X, Y ) is
continuous.
Proof.It suffices to show that the adjoint, G x TG (X, Y ) x X ! Y , of the act*
*ion
map is continuous. The map is a composition of several continuous maps.
The category GT is a closed symmetric tensor category, where S0 is the unit
object, the smash product X ^ Y is the tensor product, and the Gspace TG (X, Y*
* )
is the inner hom functor.
Define a functor GU ! GT by attaching a disjoint basepoint, X 7! X+ . This
functor is a left adjoint to the forgetful functor GT ! GU. The morphism set
GU(X, Y ) is naturally a retract of GT (X+ , Y+ ). More precisely, we have that
`
GT (X+ , Y+ ) = ZGU(Z, Y )
where the sum is over all open and closed Gsubsets Z of X. Let f :X+ ! Y+
be a map in GT . Then the corresponding unbased map is fZ :Z ! Y where
Z = X+  f1 (+). Hence, statements about based spaces often give analogous
statements for unbased spaces.
2.2. Collections of subgroups of G. We are mostly concerned with cofamilies
in this paper. For completeness, we consider more general collections of subgro*
*ups
when this is suitable.
Definition 2.3. A collection W of subgroups of G is a nonempty set of closed
subgroups of G such that if H 2 W, then gHg1 2 W for any g 2 G. A collection
W is a normal collection if for all H 2 W there exists a K 2 W such that K H
and K is a normal subgroup of G.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 7
Definition 2.4. A collection W of subgroups of G is a cofamily if K 2 W implies
that L 2 W for all subgroups L K. A collection C of subgroups of G contained
in a cofamily W is a family in W, if, for all K 2 C and H 2 W such that H K,
we have that H 2 C.
Let W be a collection of subgroups of G. The smallest cofamily of closed sub_
groups of G containing W is called the cofamily closure of W and is denoted W .
A cofamily is called a normal cofamily if it is the cofamily closure of a colle*
*ction
of normal subgroups of G.
We now give some important cofamilies.
Example 2.5. The collection of all subgroups U of G such that G=U is finite and
discrete is a cofamily. This collection of subgroups is closed under finite int*
*ersection
since G=U \ V G=U x G=V . A finite index subgroup of G has only finitely many
Gconjugate subgroups of G. Hence, if U is a finite index subgroup of G, then
\g2GgUg1 is a normal subgroup of G such that G= \g2G gUg1 is a finite discrete
group. Let fnt(G) be the collection of all normal subgroups U of G such that G=U
is a finite discrete group.
Example 2.6. Define dsc(G) to be the collection of all normal subgroups U of
G such that G=U is a discrete group. This collection is closed under intersecti*
*on.
We call a collection that is contained in the cofamily closure of dsc(G) a disc*
*rete
collection of subgroups of G.
Example 2.7. Let Lie(G) be the collection of all normal subgroups U of G such
that G=U is a compact Lie group. This collection is closed under intersection s*
*ince a
closed subgroup of a compact Lie group is a compact Lie group. We call a collec*
*tion
that is contained in the cofamily closure of Lie(G) a Lie collection of subgrou*
*ps
of G.
Lemma 2.8. Let G be a compact Hausdorff group, and let K be a closed subgroup
of G. Then {U \ K  U 2 Lie(G)} is a subset of Lie(K), and for every H 2 Lie(K)
there exists a U 2 Lie(G) such that U \ K H.
Proof.Let U 2 Lie(G). The subgroup U \K is in Lie(K) since K=K \U is a closed
subgroup of the compact Lie group G=U.
Let H be a subgroup in Lie(K). We have that \U2Lie(G)U = 1 by Corollary
A.3. Hence U \ K=H for U 2 Lie(G) is a collection of closed subgroups of the
compact Lie group K=H whose intersection is the unit element. Since Lie(G) is
closed under finite intersections, the descending chain property for closed sub*
*groups
of a compact Lie group [13, 1.25, ex. 15] gives that there exists a U 2 Lie(G) *
*such
that U \ K is contained in H.
We order fnt(G) and Lie(G) by inclusions. We recall the following facts.
Proposition 2.9. A topological group G is a profinite group precisely when
G ! limU2fnt(G)G=U
is a homeomorphism. A topological group G is a compact Hausdorff group precisely
when
G ! limU2Lie(G)G=U
is a homeomorphism.
8 H. FAUSK
Proof.These facts are wellknown. The second claim is also proved in Proposition
A.2.
Even though we are mostly interested in actions by profinite groups, we find *
*it
natural to study actions by compact Hausdorff groups whenever possible.
2.3. Model structures on the category of Gspaces. We associate to a col
lection W of closed subgroups of G a model structure on the category of based
Gspaces.
Definition 2.10. Let f :X ! Y be a map in GT . The map f is said to be
a Wequivalence if the underlying unbased maps fU :XU ! Y U are weak
equivalences for all U 2 W.
Definition 2.11. Let p: E ! B be a map in GT . We say that p is a Wfibration
if the underlying unbased maps pU :EU ! BU are Serre fibrations for all U 2 W.
We next define the generating cofibrations and generating acyclic cofibration*
*s.
We use the conventions that S1 is the empty set and D0 is a point.
Definition 2.12. Let WI be the set of maps
{(G=U x Sn1)+ ! (G=U x Dn)+ },
for n 0 and U 2 W. Let WJ be the set of maps
{(G=U x Dn)+ ! (G=U x Dn x [0, 1])+ },
for n 0 and U 2 W.
Recall that an object X in a category K is said to be a small object in K if
` `
a2A K(X, Ya) ! K(X, a Ya)
is an isomorphism for any indexing set A and any objects Ya in K.
Lemma 2.13. Let H be a closed subgroup of G, and let Y be a Hspace. Then
(G xH Y )+ is a small object in GT whenever Y+ is a small object in HT .
Proof.The result follows from the adjunctions
GT ((G xH Y )+ , Z) ~=HU(Y, Z) ~=HT (Y+ , ZH).
The restriction map GT ! HT respects arbitrary (wedge) sums.
The following model structure is called the Wmodel structure on GT . For
the definition of relative cell complexes see [25, 10.5].
Proposition 2.14. There is a proper model structure on GT with weak equiva
lences Wweak equivalences, fibrations Wfibrations, and cofibrations retracts *
*of
relative Wcell complexes. The set WI is a set of generating cofibrations and WJ
is a set of generating acyclic cofibrations.
Proof.A map p: E ! B in GT is a Wfibration if and only if it has the right
lifting property with respect to all maps in WJ. A map f is a Wacyclic fibrati*
*on
if and only if it has the right lifting property with respect to all maps in WI.
This follows from the corresponding nonequivariant result and by the fixed poi*
*nt
adjunction [26, 2.4].
To use the small object argument we need that GT (G=U+ ^ Sn1, ) commutes
with directed colimit of spaces obtained by adjoining cells in WI and WJ. This
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 9
follows from Lemma 2.13. The verifications of the model structure axioms follows
as in [26, 2.4]. The model structure is both left and right proper. This follow*
*s from
the corresponding nonequivariant results since pullbacks commute with fixed po*
*ints
and since pushouts along closed inclusions also commute with fixed points.
An alternative way to set up the model structure on GT is given in [35, III.1*
*].
Let WGT , or simply WT , denote GT with the Wmodel structure, and let
Ho(WGT ) denote its homotopy category.
Proposition 2.15. Let X be a retract of a WIcell complex, and let Y be a
Gspace. Then the set Ho(WT )(X, Y ) is isomorphic to the set of ordinary (base*
*d)
Ghomotopy classes of maps from X to Y .
Proof.All objects are fibrant and a retract of a WIcell complex is cofibrant i*
*n the
Wmodel structure. The cylinder object of (a cofibrant object) X in the Wmodel
structures is X ^ [0, 1]+ .
The next result has also been proved by Bill Dwyer [14, 4.1]. Note that a Gc*
*ell
complex X is a WIcell complex if and only if all its isotropy groups are in W.
Corollary 2.16. Let X and Y be WIcell complexes. If a map f :X ! Y is a
Wweak equivalence, then f is a Ghomotopy equivalence.
To get a topological model structure on our model category we need some as
sumptions on the collection W.
Definition 2.17. A collection W of subgroups of G is called an Illman collection
if (G=U x G=U0)+ is a WIcell complex for any two U and U0 in W.
In particular, all Illmancollections are closed under intersections since U \*
* U0 is
an isotropy group of G=U x G=U0.
Lemma 2.18. If W is a discrete or a Lie collection of subgroups of G and W is
closed under intersection, then W is an Illmancollection of subgroups of G.
______
Proof.The_statement is clear when W is contained in dsc(G). When W is contained
in Lie(G), then the claim follows from a result of Illman [29].
Lemma 2.19. Let W be an Illman collection. If X and Y are two WIcell
complexes, then X ^ Y is again (homeomorphic to) a WIcell complex.
Proof.It suffices to show that
(Sn1 x Sm1 x G=U x G=U0 ! Dn x Dm x G=U x G=U0)+
is a relative WIcell complex. Since W is an Illmancollection this reduces to
showing that
(Sn1 x Sm1 x Sk1 x G=U ! Dn x Dm x Dk x G=U)+
is a relative WIcell complex. This is so.
We follow the treatment of a topological model structure given in [35, III.1].
Note that the Gfixed points of the mapping spaces in TG are the mapping spaces
in GT . Let MG be a category enriched in GT . Let GM be the Gfixed category of
MG . Simplicial structures are defined in [25, 9.1.1,9.1.5]. We modify the defi*
*nition
of a simplicial structure by model theoretically enriching MG in the model cate*
*gory
WT instead of the model category of simplicial sets.
10 H. FAUSK
Let i: A ! X and p: E ! B be two maps in MG . Let
MG (i*, p*): MG (X, E) ! MG (A, E) xMG(A,B) MG (X, B)
be the Gmap induced by precomposing with i and composing with p.
Definition 2.20. Let MG be enriched over GT . A model structure on GM is said
to be Wtopological if it is topological (see [25, 9.1.2]) and the following ho*
*lds:
(1) There is a tensor functor X T and a cotensor functor F (T, X) in MG , for
X 2 MG and T 2 TG , such that there are natural isomorphisms of based Gspaces
MG (X T, Y ) ~=TG (T, MG (X, Y )) ~=MG (X, F (T, Y )),
for X, Y 2 MG and T 2 TG .
(2) The map MG (i*, p*) is a Wfibration whenever i is a cofibration and p is*
* a
fibration in GM, and if i or p in addition is a weak equivalence, then MG (i*, *
*p*)
is a Wequivalence.
Remark 2.21. The Gfixed points of MG (i*, p*) is GM(i*, p*). So if {G} 2 W,
then a Wtopological model structure on GM gives a topological model structure.
We prove the pushoutproduct axiom [39, 2.1,2.3].
Lemma 2.22. Let W be an Illman collection of subgroups of G. Assume that
f :A ! B and g :X ! Y are in WI, then f g :(A ^ Y ) [A^X (B ^ X) ! B ^ Y
is a Wcofibration. Moreover, if at least one of f and g is in WJ instead of WI,
then f g is a Wacyclic cofibration.
Proof.This reduces to our assumption on W; if U and U0 are in W, then G=U x
G=U0 is a Wcell complex. See also [35, II.1.22].
Proposition 2.23. Let W be an Illmancollection of subgroups of G. Then the
model structure in Proposition 2.14 is a Wtopological model structure.
Proof.This follows from [35, III.1.151.21] and Lemma 2.22.
Remark 2.24. A based topological model category M has a canonical based sim
plicial model structure. In the topological model structure denote the mapping
space by Map (M, N), the tensor by M X, and the cotensor by F (X, M). Here
X is a based space, and M and N are objects in M. The singular simplicial set
functor, sing, is right adjoint to the geometric realization functor   . The*
* corre
sponding based simplicial mapping space is given by sing(Map (M, N)). The sim
plicial tensor and cotensor are M K and F (K, M), respectively, where K is*
* a
based simplicial set and M and N are objects in M. We use that K^L ~=K^L.
A based simplicial structure gives rise to an unbased simplicial structure. W*
*e get
a unbased simplicial structure by forgetting the basepoint in the based simplic*
*ial
mapping space, and by adding a disjoint basepoint to unbased simplicial sets in
the definition of the tensor and the cotensor. Hence we can apply results about
(unbased) simplicial model structures to a topological model category.
2.4. Some change of groups results for spaces. We describe the usual adjoint
functors related to change of groups. Let K be a closed subgroup of G. The forg*
*etful
functor from GT to KT is given by restricting the Gaction to K. It has a left
adjoint given by sending X to G+ ^K X and a right adjoint given by sending X to
TK (G+ , X). Let N be a normal subgroup of G. A functor from G=Nspaces to
Gspaces is induced by the quotient map G ! G=N. The Nfixed point functor is
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 11
a right adjoint functor, and the Norbit functor is a left adjoint functor. In *
*general
these six functors do not behave well with respect to the model structures on t*
*he
categories of Gspaces, G=Nspaces, and Kspaces. We give some conditions on
the collections of subgroups of G, G=N, and K that guarantee that we get Quillen
adjunctions.
Let WK be a collection of subgroups of K, let WG be a collection of subgroups*
* of
G, and let WG=N be a collection of subgroups of G=N. The forgetful functor from
WG GT to WK KT is a Quillen right adjoint functor if WK is contained in WG . It
is a Quillen left adjoint functor if, in addition, H \ K 2 WK for every H 2 WG .
The functor from WG=N G=NT to WG GT is a right Quillen adjoint functor if
{HN=N  H 2 WG } WG=N .
It is a left Quillen adjoint functor if, in addition,
{HN  HN=N 2 WG=N } WG .
______ ______
Example 2.25. The forgetful functor from Lie(G)GT_to_Lie(K)KT is both a left
and a right Quillen adjoint functor if K_is_in_Lie(G). It is neither a left nor*
* a right
Quillen adjoint functor if K is not in Lie(G). _________
__Let_N be a normal subgroup of G. Then the functor from Lie(G=N)G=NT to
Lie(G)GT is both a left and a right Quillen adjoint functor.
3. Orthogonal GSpectra
Equivariant orthogonal spectra for compact Lie groups was introduced by Man
dell and May in [35]. We generalize their theory to allow more general groups. *
*We
develop the theory with minimal assumptions on the collection of subgroups used
to define cofibrations and weak equivalences. We follow Chapters 2 and 3 of the*
*ir
work closely.
3.1. JGVspaces. We define universes of Grepresentations.
Definition 3.1. A Guniverse U is a countable infinite direct sum 1i=1U0 of
a real Ginner product space U0 satisfying the following: (1) the onedimension*
*al
trivial Grepresentation is contained in U0; (2) U is topologized as the union *
*of all
finite dimensional Gsubspaces of U (each with the norm topology); and (3) the
Gaction on all finite dimensional Gsubspaces V of U factors through a compact
Lie group quotient of G.
If G is a compact Hausdorff group, then the Gaction on a finite dimensional
Grepresentation factors through a compact Lie group quotient of G by Lemma
A.1. This is not true in general (consider the representation Q=Z < S1). We only
use the finite dimensional Gsubspaces of U, so one might as well assume that U0
is a union of such.
Definition 3.2. Let SV denote the onepoint compactification of a finite dimen
sional Grepresentation V .
The last assumption in Definition 3.1 is added to guarantee that spaces like *
*SV
have the homotopy type of a finite Gcell complex.
Definition 3.3. If the Gaction on U is trivial, then U is called a trivial uni
verse. If each finite dimensional orthogonal Grepresentations is isomorphic to*
* a
Gsubspace of U, then U is called a complete Guniverse.
12 H. FAUSK
All compact Hausdorff groups have a complete universe. However, it might not
be possible to find a complete universes with a countable dimension. Traditiona*
*lly,
the universes have been assumed to have countable dimension [37, IX.2.1].
Remark 3.4. There are alternative notions of a Guniverses. We use the or
thogonal finite dimensional Grepresentations, that factor through a compact Lie
quotient group of G, as the indexing representations. This suffices to construc*
*t a
sensible equivariant homotopy theory for compact Hausdorff groups with the weak
equivalences determined by the cofamily closure of Lie(G).
We recall some definitions from [35, II].
Definition 3.5. Let U be a universe. An indexing representation is a finite
dimensional Ginner product subspace of U. If V and W are two indexing repre
sentations and V W , then the orthogonal complement of V in W is denoted by
W  V . The collection of all real Ginner product spaces that are isomorphic to
an indexing representation in U is denoted V(U).
When U is understood, we write V instead of V(U) to make the notation simpler.
Definition 3.6. Let JGV be the unbased topological category with objects V 2
V and morphisms linear isometric isomorphisms. Let GJ V denote the Gfixed
category (JGV)G .
Definition 3.7. A continuous Gfunctor X :JGV! TG is called a JGVspace.
(The induced map on hom spaces is a continuous unbased Gmap.) Denote the cat
egory of JGVspaces and (enriched) natural transformations by JGVT . Let GJ VT
denote the Gfixed category (JGVT )G .
Definition 3.8. Let SVG:JGV! TG be the JGVspace defined by sending V to the
one point compactification SV of V .
The external smash product
__^:J V V V V
G T x JG T ! (JG x JG )T
is defined to be X__^Y (V, W ) = X(V ) ^ Y (W ) for X, Y 2 JGV and V, W 2 V.
The direct sum of finite dimensional real Ginner product spaces gives JGV the
structure of a symmetric tensor category. A topological left Kan extension giv*
*es
an internal smash product on JGVT [36, 21.4, 21.6]. We give an explicit descrip*
*tion
of the smash product. Let W be a real Ndimensional Grepresentation in V(U).
Choose Grepresentations Vn of dimension n and Vn0of dimension N  n in V(U)
for n = 0, 1, . .,.N. For example let Vn = VN0n be the trivial ndimensional
Grepresentation, Rn. Then we have a canonical equivalence
X ^ Y (W ) ~=_Nn=0JGV(W, Vn VN0n) ^O(Vn)xO(VN0n)X(Vn) ^ Y (VN0n).
The inner hom from X to Y is the JGVspace
V 7! JGVT (X(), Y (V ))
given by the space of continuous natural transformation of JGVT functors. The
internal smash product and the inner hom functor give JGVT the structure of a
closed symmetric tensor category [35, II.3.1,3.2]. The unit object is the func*
*tor
that sends the indexing representation V to S0 when V = 0, and to a point when
V 6= 0. By passing to fixed points we also get a closed symmetric tensor struct*
*ure
on GJ VT .
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 13
3.2. Orthogonal Rmodules. For the definition of monoids and modules over
a monoid in tensor categories see [34, VII.3 and 4]. The functor SVGis a strong
symmetric functor. Hence the JGVspace SV is naturally a symmetric monoid in
GJ VT . The following definition is from [35, II.2.6].
Definition 3.9. An orthogonal Gspectrum X is a JGVspace X :JGV! TG
together with a right module structure over the symmetric monoid SV in GJGVT .
Denote the category of Gspectra by JGVS. Let GJ VS be the Gfixed category
(JGVS)G .
The smash product and inner hom functors of orthogonal spectra are the smash
product and inner hom functors of SV modules, respectively. So the category of
orthogonal Gspectra JGVS is itself a closed symmetric tensor category with SV *
*as
the unit object [35, 3.9]. The fixed point category GJ VS inherits a closed ten*
*sor
structure from JGVS. Explicit formulas for the tensor and innerhom functors are
obtained from the formulas after Definition 3.8 and [35, II.3.9].
Definition 3.10. We call a monoid R in GJ VS an algebra. We say that R is a
commutative algebra, or simply a ring, if it is a symmetric monoid in GJ VS. We
sometimes add: orthogonal, G, and spectrum, to avoid confusion.
Let R be an orthogonal algebra spectrum.
Definition 3.11. An Rmodule is a left Rmodule in the category of orthogonal
spectra. Let MVRdenote the category of Rmodules.
The category of Rmodules is bicomplete. If R is a commutative monoid, then
the category MR is a closed symmetric tensor category [35, III.7]. A monoid
T in the category of Rmodules is called an Ralgebra. Any Ralgebra is an
Salgebra.
Let T be an Ralgebra. Then the category of T modules, in the category
of Rmodules, is equivalent to the category of T modules, in the category of
Smodules, when T is regarded as an Salgebra.
We now give a pair of adjoint functors between orthogonal Gspectra and
Gspaces. The V evaluation functor
V :JGVS ! TG
is given by X 7! X(V ). We abuse language and let V also denote the functor
V precomposed with the forgetful functor from Rmodules to orthogonal spectra.
There is a left adjoint, denoted RV, of the V evaluation functor in the categ*
*ory
of Rmodules. The Rmodule RVZ, for a Gspace Z, sends W 2 V(U) to
(3.12) RVZ(W ) = Z ^ O(W )+ ^O(WV )R(W  V )
when V W , and to a point otherwise [36, 4.4]. This functor is called the V *
*shift
desuspension spectrum functor and is also denoted FV and 1V(when R = S)
in [35]. When V = 0 we denote this functor by 1R. We have that RVZ ~= SVZ^R.
3.3. Fixed point and orbit spectra. We define fixed point and the orbit spectra.
The details on adjunction functors and change of universes from [35, V] extends*
* to
our setting. The results on Quillen adjoint functors between the model structur*
*es
(constructed in later sections) require some assumptions like the ones given in
Subsection 2.4. We do not make those results explicit.
14 H. FAUSK
Let X be an orthogonal spectrum and let H be a subgroup of G. Then the
quotient X=H is defined to be X=H(V ) = X(V )=H with structure maps
X=H(V ) ^ SW ! X(V )=H ^ SW =H ~=(X(V ) ^ SW )=H ! X(V W )=H.
The Horbit spectrum is a Gspectrum with trivial Haction.
Let H be a closed subgroup of G. Let X be a Uspectrum where U is a uni
verse with trivial Haction. Then the Hfixed point spectrum XH is defined by
XH (V ) = (X(V ))H for V 2 V(U), and the structure map is
XH (V ) ^ SW ~=(X(V ) ^ SW )H ! XH (V W ).
This is a NG Hspectrum. One can also define geometric fixed point spectra as in
[35, V.5].
3.4. Examples of orthogonal GSpectra. Let T :TG ! TG be a continuous
Gfunctor. Then we define the corresponding JGVspace by T O SVG:V(G) ! TG .
This JGVspace is given an orthogonal Gspectrum structure by letting T (SV ) ^
SW ! T (SV W ) ~=T (SV ^ SW ) be the adjoint of the map
SW ! TG (SV , SV ^ SW ) T!TG (T (SV ), T (SV ^ SW ))
where the first map is a Gmap adjoint to the identity on SV ^ SW .
We can define a Gequivariant Ktheory spectrum for a compact Hausdorff
group G. If X is a compact Gspace, then KG (X) is the Grothendieck construction
on the semiring of isomorphism classes of finitely generated real bundles on X.*
* The
AtiyahSegal completion theorem generalizes to compact Hausdorff groups if we
make use of a suitable completion functor [18].
Let G be a compact Hausdorff group. We define a Thom spectrum as T OG (V ) =
colimUT OG=U (V ) where the limit is over U 2 Lie(G) such that V has a trivial
Uaction. For more detail see [18, 7].
3.5. The levelwise Wmodel structures on orthogonal GSpectra. We
make some minor modifications to the discussion of model structures in [35, III*
*].
Throughout this subsection we work in the category of Rmodules MR for a ring
R.
The category of Rmodules can be described as the category of continuous
Dspaces for an appropriate diagram category D. The objects are the same as
those of JGV, but the morphisms are more elaborate. See [36, sec.23] and [35, I*
*I.4]
for details. Interpreted as a continuous diagram category in GT , we give MR
the projective model structure inherited from the Wmodel structure on GT [25,
11.3.2].
Definition 3.13. Let 1RWI denote the collection of RVi, for all i 2 W I and
all indexing representations V in U. Let 1RWJ denote the collection of RVj, f*
*or
all j 2 WJ and all indexing representation V in U.
We call the following model structure on orthogonal Gspectra the levelwise
Wmodel structure.
Proposition 3.14. The category of Rmodules has a compactly generated proper
model structure with levelwise Wweak equivalences and levelwise Wfibrations
(as JGVdiagrams). The cofibrations are generated by 1RWI, and the acyclic
cofibrations are generated by 1RWJ. If W is an Illmancollection, then the model
structure is Wtopological.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 15
Proof.The proof of the first part is similar to [36, 6.5]. The adjunction betwe*
*en
RVand V , gives that the maps in the classes 1RWI and 1RWJ are Hurewitz
cofibrations (satisfies the homotopy extension property). Hence relative Wcell
complexes are Hurewitz cofibrations. Since W is an Illmancollection the cofibra*
*tion
hypothesis [35, III.2.6] holds. The results of Theorem 2.7 in [35, III] extends*
* if we
replace based GCW complexes with based WCW complexes.
Definition 3.15. A spectrum X is called a W  spectrum if the adjoint of
the structure maps, X(V ) ! WV X(W ) are unbased Wequivalences of spaces
for all pairs V W in V(U).
4.The stable Wmodel structure on orthogonal Gspectra
We define stable equivalences between orthogonal Gspectra [35, III.3.2].
Definition 4.1. The nth homotopy group of an orthogonal Gspectrum X at
a subgroup H of G is
ssHn(X) = colimVssHn( V X(V ))
for n 0, and
n
ssHn(X) = colimV RnssH0( V R X(V ))
for n 0, where the colimit is over indexing representations in U. A map f :X *
*! Y
of orthogonal Gspectra is a stable Wequivalence if ssHn(f) is an isomorphism
for all H 2 W and all n 2 Z.
We show in Theorem 4.7 below that we can Bousfield localize the levelwise
model structure on the category of orthogonal Gspectra with respect to the sta*
*ble
Wequivalences.
We follow our program of giving model structures to the category of orthogonal
Gspectra with minimal assumptions on the collection of subgroups used. We
need to strengthen the notion of an Illmancollection of subgroups of G. The ext*
*ra
condition added is needed in the proof of Corollary 4.4 and Proposition 4.5.
Definition 4.2. Given a Guniverse U. We say that a collection C is a UIllman
collection if C is an Illmancollection (see Def. 2.17), and if in addition the *
*following
holds: whenever H is the stabilizer of and element in the universe U, then H \ U
is in C for all U 2 C.
If U is a trivial Guniverse, then a collection is UIllmanif it is Illman. I*
*f U is
a complete universe, then C is a UIllmancollection if C is a family in the cof*
*amily
closure of Lie(G) (see Def. 2.4).
4.1. Verifying the model structure axioms.
Lemma 4.3. Let X, Y , and Z be based Gspaces. Let W be an Illmancollection.
If Z is a Wcell complex and f :X ! Y is a Wequivalence, then TG (Z, X) !
TG (Z, Y ) is a Wequivalence.
Proof.The higher lim spectral sequence shows that is enough to prove the result
when K is Sn+^ G=L+ for L 2 W. An adjunction gives that
ssHk(TG (Sn+^ G=L+ , X)) ~=[Sn+^ Sk ^ (G=L x G=H)+ , X]G ,
16 H. FAUSK
where the square brackets are Ghomotopy classes of maps (see Prop. 2.15). Since
Sn+^ Sk is a based CWcomplex and (G=L x G=H)+ is a based Wcell complex
by the definition of an Illmancollection in 2.17, the result follows.
Corollary 4.4. Let W be a UIllman collection of subgroups of G. A levelwise
Wequivalence of Gspectra is a stable Wequivalence.
Proof.We have assumed that any finite dimensional Grepresentation V in the
universe U is a G=Urepresentation for a compact Lie group quotient G=U of G.
Since the collection W is UIllman, it then follows that SV is a (finite) Wcell
complex. Lemma 4.3 gives that V f(V 0) is a Wequivalence for all V, V 02 V.
Each ssU*, for U 2 W, is a homology theory on the homotopy category of orthog
onal Gspectra with the levelwise Wmodel structure. This follows by Corollary
4.4 since the Hfixed point functor commutes with wedges and with pushout along
a closed inclusion (all Hurewicz cofibrations are closed inclusions since our s*
*paces
are weak Hausdorff). We get a stable Wmodel structure on orthogonal Gspectra
by Bousfield localizing the levelwiseLWmodel category of orthogonal spectra wi*
*th
respect to the homology theory h = U2W ssU*.
We give a more precise description of this stable Wmodel structure, and de
termine the hlocal objects. We follow [35, III.4]. A set of generating cofibra*
*tions
is 1RWI. We give a set of generating acyclic cofibrations. Let
~V,W : RV WSW ! RVS0
be the adjoint of the map
SW ! ( RVS0)(V W ) ~=O(V W )+ ^O(W) R(W )
given by sending an element w in SW to e ^ i(W )(w) where e is the identity ma*
*p in
O(V W ), and i: S0 ! R is the unit map. Let kV,W be the map from RV WSW
to the mapping cone, M~V,W, of ~V,W. Let WK be the union of 1RWJ and the
set of maps of the form i kV,W for i 2 WI and indexing representations V, W in
U. The box is the pushoutproduct map. The set WK of maps in MR is a set of
generating acyclic cofibrations.
Note that if W is a UIllmancollection of subgroups of G, A is a based Wcell
complex, and V is an indexing representation in U, then A ^ SV is again a based
Wcell complex by Lemma 2.19. The next result, together with Corollary 4.4,
show that a map between  Wspectra is a levelwise Wequivalence if and only
if it is a Wequivalence. This fundamental result is an extension of [35, III.9*
*].
Proposition 4.5. Assume that W is a UIllmancollection of subgroups of G. Let
f :X ! Y be a map of W   Gspectra. If
f*: ssH*(X) ! ssH*(Y )
is an isomorphism for any H 2 W, then for all indexing representations V U
f(V )*: ssH*(X(V )) ! ssH*(Y (V ))
is an isomorphism for all H 2 W. So f is a level Wequivalence.
Proof.Let Z be the homotopy fiber of f. It is again an  Gspectrum. We
want to show that ssH*(Z) = 0 for all H 2 W, implies that ssH*(Z(V )) = 0 for
any indexing representations V and any H 2 W. Fix an indexing representation
V and a normal subgroup N 2 Lie(G), such that N acts trivially on V . With
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 17
these choices ( V Z(V ))H = V (Z(V )H ) for all H N. Hence ssH*+V(Z(V )) *
*is
isomorphic to ssH*( V Z(V )) for all H N in W. Since Z is an  Gspectrum,
an easy argument gives that ssH*(Z(V )) = 0 for all H 2 W such that H N [35,
III.9.1].
We now prove the result for subgroups H in W that are not necessarily contain*
*ed
in N. Fix a subgroup H 2 W. Assume by induction that ssK*(Z(V )) = 0 for all
subgroups K 2 W such that K H. If L is an orbit type in V , then H \ gLg1
is in W for all g 2 G since C is a UIllmancollection. The argument given in [3*
*5,
III.9] implies that ssH*(Z(V )) = 0. We now justify that we can make the induct*
*ive
argument. The quotient group H=H \ N is isomorphic to H . N=N, which is a
subgroup of the compact Lie group G=N. Hence the partially ordered set of closed
subgroups of H containing H \ N satisfies the descending chain property. We have
that ssK*(Z(V )) = 0 for all K H \ N in W. We start the induction with the
subgroup H \ N which by assumption is in W. For more details see [35, III.9].
As in [35, III.4.8] and [36, 9.5], the fact that the Wmodel structure on G *
* T
is Wtopological gives the following characterization of the maps that satisfy *
*the
right lifting property with respect to WK.
Proposition 4.6. A map p: E ! B satisfies the right lifting property with respe*
*ct
to WK if and only if p is a levelwise fibration and the obvious map from E(V ) *
*to
the pullback of the diagram
W E(V W )


fflffl
B(V )_____// W B(V W )
is an unbased Wequivalence of spaces for all V, W 2 V(U).
Proof.The only modification of the proof in [35, III.4.8] is that we now have a
topological model category such that if i is a Wcofibration, and p is a Wfibr*
*ation
then (MR )G (i*, p*) is a Wfibration of spaces.
Theorem 4.7. Let W be a UIllmancollection of subgroups of G. Let R be a ring.
The category of Rmodules is a compactly generated proper Wtopological tensor
model category such that the weak equivalences are the stable Wequivalences, t*
*he
cofibrations are retracts of relative 1RWIcell complexes, and the acyclic cof*
*ibra
tions are retracts of relative WKcell complexes.
Proof.The proof is almost identical to the proofs in [35, III.4],[35, III.7.4],*
* and [36,
9]. Note that the proofs uses a few lemmas, given in [35], that are not explici*
*t in
this paper. More details of the tensor structure are given in Lemma 5.9.
We sometimes denote MR together with the Wmodel structure by WMR .
Lemma 4.8. Let H be in W. The stable homotopy group ssHn is corepresented
by RRnG=H+ , for n 0, and by R G=H+ ^ Sn, for n 0, in the category
of Rmodules. The homotopy group ssHn is a homology theory which satisfies the
colimit axiom.
The colimit axiom says that colimassH*(Xa) ! ssH*(X) is an isomorphism,
where the colimit is over all finite subcomplexes Xa of the cell complex X.
18 H. FAUSK
4.2. Fibrations. We summarize the description of the fibrations in the stable
Wmodel structure.
Proposition 4.9. A map f :X ! Y is a fibration if and only if the map X(V ) !
Y (V ) is a Wfibration and the obvious map from X(V ) to the pullback of the
diagram
W X(V W )


fflffl
Y (V )____// W Y (V W )
is a unbased Wequivalence of spaces, for all V, W 2 V. The fibrant spectra are
exactly the W  spectra. A map f :X ! Y is an acyclic fibration if and only if
f is a levelwise acyclic fibration.
A natural fibrant replacement functor in MR is given by sending an Rmodule
X to the Rmodule
(4.10) V 7! colimW W X(V W )
where the colimit is over indexing representations in U. In particular, a natu*
*ral
fibrant replacement of the suspension spectrum RVZ (see 3.12) is the spectrum
that sends V 0to
colimW W+V O(V V 0 W ) ^O(V 0 W)R(V 0 W ) ^ Z.
The next Lemma (and the claim after it) follows as in [35, 3.6, 3.11].
Lemma 4.11. Let f :X ! Y be a map of spectra and let V 2 V. Then X ^ SV !
Y ^ SV is a stable Wequivalence if and only if f is a stable Wequivalence.
More generally, X ^ A ! Y ^ A is a Wequivalence for all Wcell complexes
A and Wequivalences X ! Y .
Definition 4.12. Let W be a collection of subgroups of G, and let K be a subgro*
*up
of G. The intersection K\W is defined to be the collection of all subgroups H 2*
* W
such that H K.
If W is a UIllmancollection of subgroups of G and K 2 W, then K \ W is a
UKIllmancollection of subgroups of G.
Lemma 4.13. Let W be a UIllmancollection of subgroups of G and let K 2 W.
Let Y be a fibrant object in W  GMR . Then Y regarded as a Kspectrum is
fibrant in (K \ W)  K MR .
Proof.This follows from the explicit description of fibrant objects in Proposit*
*ion
4.9. (Alternatively, check that G^K  is left Quillen adjoint to the forgetful *
*functor
from Gspectra to Kspectra.)
Lemma 4.13 need not remain true when the subgroup K is not in W. For
applications in Section 10 we give some assumptions that guarantee that the res*
*ult
remains true even when K 62 W.
Lemma 4.14. Let W be a UIllmancollection of subgroups of a compact Hausdorff
group G. Let f :X ! Y be a fibration in WMR . Assume that both X and Y
are W  Scell complexes. Let K be any closed subgroup of G, and let W0 be a
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 19
UKIllmancollection of subgroups of K such that W0W W. Then f, regarded
as a map of Kspectra, is a fibration in the W0model structure on KMR (with
the universe UK, and R regarded as a Kspectrum).
Note that X and Y are required to be W Scell complexes not just W Rcell
complexes. This holds if they are W  Rcell complexes and R is a W  Scell
complex.
Proof.Let f :X ! Y be a Wfibration between Wcofibrant objects in WGMR .
Since W0 is UKIllmanas a Kcollection of map, it suffices, by Proposition 4.9,
to show that for any L 2 W0 the map f(V )L :X(V )L ! Y (V )L is a fibration, for
V 2 V, and to show that the map from X(V )L to the pullback of the diagram
(4.15) Y (V )L


fflffl
( W X(V W ))L_____//( W Y (V W ))L
is an equivalence of spaces, for V, W 2 V.
We need that maps from a compact space C to the Lfixed points of X(V ) and
Y (V ) factor through the ULfixed points of X(V ) and Y (V ) for some U 2 W.
Since X and Y are W  Scell complex. A map from a compact space C into a
W  Scell complex factors through a finite sub cell complex. Hence it suffices*
* to
verify the claim for individual cells.
Let Z be a Wcell complex space. Then a map from a compact space C into
V 0Z(V ) factors through ( V 0(V )Z)U for some U 2 W. We prove this claim.
Note that O(W ) and G=H are Wcell complex for every indexing representation
W and every H 2 W, respectively0[29]. Recall, from 3.12, that V 0(V ) is the s*
*pace
Z ^ O(V )+ ^O(V V 0)SV V , for V V 0, and a point otherwise. A map from a
compact space C into the quotient of a Wcell complex Z0divided out by a compact
group action, lies in the quotient of a compact subset of Z0. The claim follows.
Hence a map from a compact space C into the Lfixed points of X(V ) and
Y (V ) factor through X(V )UL and Y (V )UL , respectively, for some U 2 W.
We are now ready to prove the Lemma. Let
(4.16) Dn+ _______//_X(V )L
j  f(V)L
fflffl fflffl
(Dn x I)+ ____//_Y (V )L
be a diagram of based spaces. There exists a U 2 W such that the map from j
to f(V )L factors through f(V )UL . Since f(V )UL is a fibration we get a lift *
*in the
diagram 4.16. Hence f(V )L is a fibration.
The proof of the second claim is similar. We note that a map from a compact
space C to ( W X(V W ))L, composed with the inclusion into W X(V W ), is
adjoint to0a based map from C+ ^ SW to X(V W ). Hence it factors through
X(V W )U , for some U0 2 W. By choosing a smaller U U0 such that U acts
trivially on W , the map from C factors through ( W X(V W ))UL . Hence to check
that the map from X(V )L to the pullback of 4.15 is a weak equivalence, it suff*
*ices to
check this on all ULfixed points for U 2 W. This follows by our assumptions.
20 H. FAUSK
Lemma 4.17. Let W be a UIllmancollection of subgroups of a compact Hausdorff
group G. Let f :X ! Y be a (co)nequivalence in MR between fibrant objects X
and Y in the Wmodel structure on GMR , that are also W  Scell complexes.
Let K be any closed subgroup of G, and let W0 be a UKIllman collection of
subgroups of K such that W0W W. Then f regarded as a map of Kspectra is
a (co)nequivalence in the W0model structure on KMR .
Proof.Both X and Y are fibrant, so
X(V ) ! Y (V )
is a W(co)nequivalence, for every indexing representation V (by a modificati*
*on
of the proof of 4.14). Since X and Y are cofibrant we get, as in the proof of L*
*emma
4.14, that X(V )L ! Y (V )L is a (co)nequivalence for any L 2 W0.
4.3. Positive model structures. We give some brief remarks about other model
categories of spectra. Prespectra are defined by replacing the category V(U), *
*in
Definitions 3.6 and 3.9, by a smaller category consisting of the indexing repre*
*sen
tations and the inclusions. There is a stable Wmodel structure on the category
of prespectra. This model category is Quillen equivalent to the stable Wmodel
structure on Gorthogonal spectra [35, III.4.16].
We can also consider model structures on the category of algebras. We need
to remove some of the cofibrant and acyclic cofibrant generators to make sure t*
*he
free symmetric algebra construction takes acyclic cofibrant generators to stable
Wequivalences. Let R+WI and R+WJ consist of all V desuspensions of ele
ments in WI and WJ by indexing representations V in U such that V G 6= 0.
The positive levelwise Wmodel structure on the category of orthogonal spectra
is the model structure obtained by replacing 1RWI and 1RWJ by R+WI and
R+WJ, respectively. The positive stable Wmodel structure on orthogonal spec
tra is obtained by replacing WK by the set WK+ consisting of the union of R+J
and the maps i kV,W with i 2 I and V G 6= 0. The discussion of the positive mod*
*el
structure goes through as in [35, III.5].
Proposition 4.18. Let R be a commutative monoid in the category of Gorthogonal
spectra. Then there is a compactly generated Wtopological model structure on t*
*he
category of Ralgebras such that the fibrations and weak equivalences are creat*
*ed
in the underlying positive Wmodel category of orthogonal Gspectra. The same
applies to the category of commutative Ralgebras.
4.4. Homotopy classes of maps between suspension spectra. We first give
a concrete description of the set of morphisms between suspension spectra in the
Wstable homotopy category on MR . We then prove some results about vanishing
of the negative stable stems; they are used in Section 7.
Recall that WT denote GT with the Wmodel structure.
Lemma 4.19. Let X and Y be two based Gspaces. Then there is a natural
isomorphism
Ho(WMR )( 1RX, 1RY ) ~=Ho(WT )(X, colimW W (R(W ) ^ Yc)),
where Yc is a cofibrant replacement of Y .
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 21
Proof.Recall the description of 1Rin 3.12. The functors 1Rand 0 are a Quillen
adjoint pair. The result follows by replacing X and Y by cofibrant objects, Xc *
*and
Yc, in WT , and then replace 1RYc by a fibrant object as in 4.10.
Corollary 4.20. Let X and Y be two based Gspaces. Then there is a natural
isomorphism
Ho(WMS)( 1 X, 1 Y ) ~=Ho(WT )(X, colimW W SW Y ).
In particular, if X is a finite Wcell complex, then
Ho(WMS)( 1 X, 1 Y ) ~=colimWHo(WT )(X ^ SW , Y ^ SW ).
We next show that the negative stable stems are zero. In what follows homotopy
means usual homotopy (a path in the space of maps).
Lemma 4.21. Let V be a finite dimensional real Grepresentation, with Gaction
factoring through a Lie group quotient of G. Let X be a based Gspace, and let
n > 0 be an integer. Then any based Gmap
SV ! SV ^ X ^ Sn
is Gnullhomotopic.
Proof.Assume the action on SV factors through the compact Lie group quotient
G=K. The problem reduces to show that SV ! SV ^ XK ^ Sn is G=Knull
homotopic for all n > 0. Hence we can assume that G is a compact Lie group.
By Illman's triangulation theorem SV is a finite Gcell complex [29]. We choose*
* a
GCW structure on SV . Let (G=HixDni)+ be a cell of SV . We take the Hifixed
points of SV and compare the real manifold dimensions, denoted dim, of the fixed
points of SV and the cells in SV . This gives that ni= dim(V Hi)  dim(NG Hi=Hi*
*).
To prove the Lemma it suffices to show that any given map f :SV ! SV ^ X ^ Sn
extends over the cone SV ^ I of SV . There is a sequence
SV = Y1 ! Y0 ! Y1 ! . .!.YN = SV ^ I
where Yn+1 is obtained from Yn by a pushout
W
G=Hi +^ Sni ______//_Yn
 
 
W fflffl fflffl
G=Hi +^ Dni+1 ____//_Yn+1
where the wedge sum is over all i such that ni= n, and N satisfies ni N for all
i. Hence it suffices to show that any map _G=Hi + ^ Sni ! SV ^ X ^ Sn is Gnull
homotopic for all i. This is equivalent to showing that Sni ! SV Hi^XHi^Sn is n*
*ull
homotopic, which is true because ni= dim(V Hi)  dim(NG Hi=Hi) < dim(V Hi) +
n.
Lemma 4.22. Let U be any Guniverse, and let W be a UIllmancollection of
subgroups of G. Then we have that
Ho(WMS)( 1 G=H+ , 1 G=K+ ^ Sn) = 0,
for all H, K 2 W and n > 0.
22 H. FAUSK
Proof.Since W is a UIllmancollection of subgroups of G, SV ^ G=H+ is a finite
Wcell complex by Lemma 2.19 and compactness of SV ^ G=H+ . Corollary 4.20
gives that the group Ho(WMS)( 1 G=H+ , 1 G=K+ ^ Sn) is isomorphic to
colimV 2V(U)Ho(WT )(SV ^ G=H+ , SV ^ G=K+ ^ Sn).
It suffices to show that any map SV ^ G=H+ ! SV ^ G=K+ ^ Sn is Gnull
homotopic. This is equivalent to show that SV ! SV ^ (G=K)+ ^ Sn is Hnull
homotopic. This follows from Lemma 4.21.
This Lemma allow us to form WCWcomplex approximation.
Lemma 4.23. Let W be a UIllman collection of subgroups of G. Let T be an
Smodule such that TjH = 0, for j < n and H 2 W. Then there is a cell complex,
T 0, built out of cells of the form 1Rk0Sk1^G=H+ ! 1Rk0Dk^G=H+ , for kk0 n
and H 2 W, and a Wweak equivalence T 0! T .
Proof.The approximation can be constructed as a WCWcomplex using Lemma
4.22.
Lemma 4.24. Let W be a UIllman collection of subgroups of G. Let R and T
be two Smodules. If RHi= 0, for i < m and H 2 W, and TjH = 0, for j < n and
H 2 W, then (R ^ T )Hk= 0 for k < m + n and H 2 W.
Proof.We can replace R and T by WIcell complexes made of cells in dimension
greater or equal to m and n, respectively by Lemma 4.23. The spectrum analogue
of Lemma 2.19 gives that R ^ T is again a Wcell complex made out of cells in
dimension greater or equal to m + n. The result now follows from Lemma 4.22.
Proposition 4.25. Let W be a UIllmancollection of subgroups of G. Let R be
a ring spectrum such that RHn= 0 for all n < 0 and H 2 W. Then we have that
Ho(WMR )( 1RG=H+ , 1RG=K+ ^ Sn) = 0,
for all H, K 2 W and n > 0.
Proof.The group Ho(WMR )( 1RG=H+ , 1RG=K+ ^ Sn) is isomorphic to
Ho(WMS)( 1 G=H+ , 1 G=K+ ^ R ^ Sn).
The result now follows from Lemma 4.21 and Proposition 4.24. (Let T be 1 G=K+ ^
Sn.)
Lemma 4.26. Let W be a UIllmancollection of subgroups of G. Let R be a ring
such that RHn= 0, whenever n < 0 and H 2 W. Let T be an Rmodule such that
TjH = 0, for j < n and H 2 W. Then there is a cell complex, T 0, built out of
cells of the form 1RSk1 ^ G=H+ ! 1RDk ^ G=H+ , for k n, and a Wweak
equivalence T 0! T .
Proof.This follows from Lemma 4.25 and the proof of Lemma 4.23.
If the universe U is trivial and K is a not subconjugated to H in G, then the*
*re
are no nontrivial maps from 1 G=H+ ^ Sn to 1 G=K+ ^ Sm . We take advantage
of this to strengthen Lemma 4.25.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 23
Proposition 4.27. Let U be a trivial universe. Let W be an Illmancollection of
subgroups of G. Let R be ring spectrum such that RHn= 0 for all n < 0 and H 2 W.
Then, for each H, K in W, we have that
Ho(WMR )( 1RG=H+ , 1RG=K+ ^ Sn) = 0,
whenever n > 0 or H is not subconjugated to K.
Proof.If n > 0, then the result follows from 4.25. If H is not subconjugated to*
* K,
then
Ho(WMR )( 1RG=H+ , 1RG=K+ ^ Sn) ~=colimm ss0 m (G=K+ ^ (R ^ Sn)(Rm ))H .
This is 0 since G=KH+is the basepoint.
4.5. The Segaltom Dieck splitting theorem. We consider homotopy groups
of suspension spectra. Let G be a_compact_Hausdorff group, let U be a complete
Guniverse, and let MS have the Lie(G)model structure.
Proposition 4.28. If Y is a Gspace, then there is an isomorphism of abelian
groups L
H ss*(EWG H+ ^WGH Ad(WGH)Y H) ! ssG*( 1SY ) ______
where the sum is over all Gconjugacy classes of subgroups H in Lie(G).
Proof.We have an isomorphism
colimN2Lie(G)ss*(colimV 2UN( V SV Y N)G ) ! ssG*( 1SY )
and UN is G=Ncomplete. The result follows from the splitting theorem for com
pact Lie groups [33, V.9.1].
If U is a complete Guniverse, then U restricted_to K is again a complete
Kuniverse [17, Sec. 3]. So for any K 2 Lie(G), the stable Khomotopy groups
of a Gspace Y calculated in the Ghomotopy category are isomorphic to those
calculated in the Khomotopy_category. Hence the calculation of the nth homo
topy group at K 2 Lie(G)of a Gspace Y reduces to Proposition 4.28 (with G
replaced by K).
4.6. Selfmaps of the unit object. The additive tensor category, Ho(WMR ), is
naturally enriched in the category of modules over the ring,
Ho(WMR )( 1RS0, 1RS0),
of self map of the unit object, 1RS0, in the homotopy category. Let us denote *
*this
ring by BRW, and denote BS0Wsimply by BW . The ring BRW depends on G, W, R,
and the Guniverse U. If G 2 W, then we can identify BRW with ssG0(R).
If A is an Ralgebra, then BAW is an BRWalgebra. Since all algebras of ortho*
*g
onal spectra are Salgebras, it is important to understand the ring BW .
If G is a compact Lie group, the universe is complete, and W is the collectio*
*n of
all closed subgroups of G, then BW is naturally isomorphic to the Burnside rin*
*g,
A(G), of G [37, XVII.2.1].
______
Lemma 4.29. Let U be a complete Guniverse and let W be the collection Lie(G).
Then the selfmaps of S0, in the homotopy category of WM, is naturally isomorph*
*ic
to
colimU2Lie(G)A(G=U),
24 H. FAUSK
where A(G=U) ~=Ho(WM)(G=U+ , S0) is the Burnside ring of the Lie group G=U
and the maps in the colimit are induced by the quotient maps G=U+ ! G=V+ , for
V < U in Lie(G).
In general, it is difficult to determine BW . For example, when G is a finite*
* group
and W is a family, then the proof of the Segal conjecture gives that the ring B*
*W is
isomorphic to the Burnside ring A(G) of G completed at the augmentation ideal
\H2W ker(A(G) ! A(H)),
where the maps A(G) ! A(H) are the restriction maps [37, XX.2.5].
We give an elementary observation which shows that different collections W
might give rise to isomorphic rings BW .
Lemma 4.30. Let N be a normal subgroup of a finite group G, and let WN be
the family of all subgroups contained in N. If X 2 GT has a trivial Gaction and
Y 2 GT , then
Ho ({N}MR )( 1 X, 1 Y ) ! Ho(WN MR )( 1 X, 1 Y )
is an isomorphism.
In particular, B{N} is isomorphic to BWN .
Proof.Let X0 be a cell complex_replacement_of X built out of cells with trivial
Gactions. The space EWN is Lie(G)equivalent to E(G=N). This is a {N}cell
complex. Hence X0^ EG=N+ ! X0 is a cofibrant replacement of X both in the
{N} and and in the WN model categories. A WN fibrant replacement Y 0of Y is
also a {N}fibrant replacement.
Remark 4.31. If X does not have trivial Gaction, then the homotopy classes
[ 1 X, 1 Y ] in Lemma 4.30 are typically different for the collections {N} and*
* WN ,
respectively.
5.The W  Cmodel structure on orthogonal Gspectra
Let R be a ring and let C be a UIllman collection of subgroups of G. We
define Kequivalences in the Cmodel structure on the category of Rmodules,
MR , for K not necessarily in C. Then we construct a model structure with weak
equivalences detected by a collection W of subgroups of G that is not necessari*
*ly
contained in C. We start by briefly describing the W  Cmodel structure on MR
in the case when W is contained in C.
Let C be a UIllmancollection of subgroups of G. Let H 2 C. Then ssH* is a
corepresented homology theory that satisfies the colimit axiom by Lemma 4.8. The
direct sum L
h = K2W,n2Z ssKn
is also a homology theory that satisfies the colimit axiom. The hequivalences
are closed under pushout along cofibrations in CMR . We can now (left) Bousfield
localize CMR with respect to the homology theory h [4] [25, 13.2.1]. Hence for
any subcollection W in C there is a model structure on Gspectra such that the
cofibrations are retracts of relative Ccell complexes and the weak equivalence*
*s are
maps f such that ssHn(f) is an isomorphism, for all H 2 W and all n 2 Z.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 25
5.1. The construction of WCMR . We define homotopy groups in the Cmodel
category with respect to subgroups not necessarily in C.
Definition 5.1. Let C be a UIllman collection_of subgroups of G. Let K be
a subgroup of G such that the closure UK 2 C, for all U 2_C._ The nth stable
homotopy group at K is defined to be K*(X) = colimU2C ssUK*(X).
The colimit is over the category with objects subgroups U of G that are in C *
*and
with morphisms containment of subgroups. The colimit is directed since C is an
Illmancollection. If K 2 C, then K*and ssK*are canonically isomorphic functors.
Definition 5.2. Let C and W be two collections of_subgroups_of G. Then the
product collection CW has elements the closure, UH , of the product subgroup UH
in G, for all U 2 C and all H 2 W.
Example 5.3. The collection W = {1} satisfies CW C for any collection C. If C
is a cofamily, then CW C for any collection W.
Definition 5.4. Let W be a collection of subgroups of G such that CW C. Then
we say that a map f between orthogonal spectra is a Wequivalence if Kn(f)
is an isomorphism for all K 2 W and all integers n.
Directed colimts of abelian groups respect direct sums and exact sequences. So
K* is a homology theory which satisfies the colimit axiom by Lemma 4.8. The
direct sum L
h = K2W,n2Z Kn
is again a homology theory which satisfies the colimit axiom. Hence we can Bous
field localize with respect to h.
Theorem 5.5. Let C be a UIllmancollection of subgroups of G, containing the
subgroup G, and let W be any collection of subgroups of G such that CW C.
Then there is a cofibrantly generated proper simplicial model structure on MR s*
*uch
that the weak equivalences are Wequivalences and the cofibrations are retracts*
* of
relative Ccell complexes.
Proof.There exists a set K of relative CGCW complexes with sources CGCW
complexes such that a map p has the right lifting property with respect to all
hacyclic cofibrations with cofibrant source, if and only if p is a fibration a*
*nd p
has the right lifting property with respect to K. To find such a set of maps K
we use the cardinality argument of Bousfield.QWe need to take into account both
the cardinality of G and the cardinality of V R(V ), where the product is over
indexing representations in the universe U. The class of hequivalences is clos*
*ed
under pushout along Ccofibrations. Hence we can apply [25, 13.2.1] to conclude
that if p has the right lifting property with respect to the maps in the set K,
then it has the right lifting property with respect to all h*acyclic cofibrati*
*ons.
Hence there is a cofibrantly generated left proper model structure on MR with t*
*he
specified class of cofibrations and weak equivalence [25, 4.1.1]. It remains to*
* show
that the model structure is right proper and simplicial.
We show that the model structure is right proper. The Ufixed point functor,
for U 2 C, takes pullback squares to pullback squares, and fibrations to fibr*
*ations.
Hence the claim follows from properness of the model category of orthogonal spe*
*ctra
[36, 9.2].
26 H. FAUSK
We show that the model structure is simplicial. This is where we use the as
sumption that G is contained in C. The tensor and cotensor functors are given by
1RK^X and F ( 1RK, X), respectively, for a simplicial set K and an Rmodule
X. The simplicial hom functor is given by singGMR (X, Y ). It is clear that the
pushoutproduct map applied to a simplicial cofibration and a Ccofibration in *
*MR
is again a Ccofibration. If the simplicial cofibration is acyclic, then the pu*
*shout
product map is in fact a Cacyclic cofibration. This follows since MR , with the
Cmodel structure, is a Csimplicial model structure. It suffices to show that*
* if
X2 ! Y2 is a W  Cacyclic cofibration with Ccofibrant source, then the map
from the the pushout of
1RSn1 ^ X2_____// 1RDn ^ X2


fflffl
1RSn1 ^ Y2
to 1RDn ^ Y2 is again a W  Cacyclic cofibration [39, 2.3]. This is the case *
*since
our weak equivalences are given by a homology theory in the homotopy category
of the tensor Cmodel structure on MR (see Theorem 4.7).
This model structure is called the W  Cmodel structure on MR . The
Wmodel structure is the W  Wmodel structure. We sometimes denote MR
together with the W  Cmodel structure by WCMR .
Proposition 5.6. Let C1 C2 be two UIllmancollections of subgroups of G and
let W be a collection of subgroups of G such that C1W C1 and C2W C2. Then
the identity functors WC1MR ! WC2MR and WC2MR ! WC1MR are left and
right Quillen adjoint functors, respectively. Hence a Quillen equivalence.
Given two UIllmancollections C1 and C2 such that C1W C1 and C2W C2.
Then the union of the two collections C = C1 [ C2 is also a UIllmancollection
such that CW C. The identity functors are left Quillen equivalences from the
C1  Wmodel structure on MR , and from the C2  Wmodel structure on MR
to the C1 [ C2  Wmodel structure on MR .
Remark 5.7. One can also construct a W  Cmodel structure on the category
of based Gspaces, GT . This is obtained by localizing with respect to the clas*
*s of
Whomotopy equivalences.
5.2. Tensor structures on MR . The category MR is a closed symmetric tensor
category. We follow [39, 2] when considering the interaction of model structur*
*es
and tensor structures. A model structure is said to be tensorial if the follow*
*ing
pushoutproduct axiom is valid.
Definition 5.8. The pushoutproduct axiom [39, 2.1]: Let f1: X1 ! Y2 and
f2: X2 ! Y2 be cofibrations. Then the map from the pushout, P, to Y1 Y2 in the
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 27
diagram
f1 1
X1 X2 _____//Y1 X2
::
1 f2 ::::
fflffl fflffl1:f2:
X1 Y2________//PL ::
VVVVV LLL ::
VVVVVV LLL ::
f 1VVVVV LLL::
1 VVV**AEAE:&&L
Y1 Y2,
is again a pushout. If, in addition, one of the maps f1 or f2 is a weak equival*
*ence,
then P ! Y1 Y2 is also a weak equivalence.
The monoid axiom [39, 2.2]: Any acyclic cofibrations tensored with arbitrary
object in M is a weak equivalence. Moreover, arbitrary pushouts and transfinite
composition of such maps are weak equivalences.
Lemma 5.9. The Wlevelwise model structure on MR satisfies the pushout
product axiom and the monoid axiom.
Proof.Since 1R is a left adjoint functor, the verification of the pushoutprod*
*uct
axiom for MR reduces to WT , which is Lemma 2.22.
It suffices to check the monoid axiom [39, 2.2]. The acyclic cofibrant genera*
*tors
are of the form RV(G=U x Dn)+ ! RV(G=U x Dn x [0, 1])+ . This is a deformation
retract. So its smash product with any spectrum X is again a deformation retrac*
*t.
Hence a Wequivalence. Pushout along a deformation retract is again a deformati*
*on
retract. The class of Wequivalences is closed under transfinite composition. H*
*ence
the class of acyclic cofibrations tensor arbitrary objects in MR are contained *
*in W,
and pushout and transfinite compositions of such maps is again in W.
Lemma 5.10. Let M, be a closed tensor category. If M is a cofibrantly generat*
*ed
model structure that satisfies the pushoutproduct axiom, then a localized model
structure on M, with the same cofibrations and a larger class of weak equivalen*
*ces,
W0, also satisfies the pushoutproduct axiom provided the following holds:
(1) the sources of the cofibrant generators of M are cofibrant;
(2) the class of cofibrations in W0 is closed under pushouts; and
(3) if X is cofibrant and f is a cofibration in W0, then X f is in W0.
Proof.The first part of the pushoutproduct axiom is immediate since the cofibr*
*a
tions are unchanged in the localized model structure.
Let f1: X1 ! Y2 be a cofibration in W0. Let f2: X2 ! Y2 be a cofibrant
generator. Consider the diagram
f1 1
X1 X2 _____//Y1 X2
::
1 f2 ::::
fflffl fflffl1:f2:
X1 Y2________//PL ::
VVVVV LLL ::
VVVVVV LLL ::
f 1VVVVV LLL::
1 VVV**AEAE:&&L
Y1 Y2,
28 H. FAUSK
where P is the pushout. The first and third conditions give that f1 1X2 and
f1 1Y2 are cofibrations in W0. The second condition gives that X1 Y2 ! P is
in W0. The two out of three axiom now gives that P ! Y1 Y2 is in W0.
Proposition 5.11. The tensor (closed) category MR , with the WCmodel struc
ture, satisfies the pushoutproduct axiom. Hence it is a tensor model category.
Proof.This follows from Lemmas 5.9 and 5.10. The third condition in the Lemma
reduces to cell level considerations. See the proof of Theorem 5.5.
Remark 5.12. In fact, Remark 5.7 and Proposition 5.11 give that the W 
Cmodel structure on MR is a W  Ctopological model structure.
Proposition 5.13. Let U be a complete Guniverse. Let W be a UIllmanLie col
lection of subgroups of G. Then the dualizable objects in WMR (with the Wmodel
structure) are precisely retracts of V(U)desuspensions of finite Wcell comple*
*xes.
Proof.The proof in [37, XVI 7.4] goes through with modifications to allow for
general Rmodules instead of S0modules.
5.3. The Ccofree model structure on MR . The W  Cmodel structure on
MR is of particular interest when W = {1}.
Definition 5.14. We say that f is a Cunderlying equivalence if {1}(f) =
colimU2CssU (f) is an equivalence.
The name, Cunderlying equivalence, is justified by the next lemma.
Lemma 5.15._Assume_that G is a compact Hausdorff group and let C be the col
lection Lie(G). Let R be a nonequivariant ring spectrum. Then a map f :X ! Y
between cofibrant objects (retracts of Ccell complexes) is a Cunderlying equi*
*va
lence if and only if f is a nonequivariant weak equivalence.
Proof.This follows as in the proof of Lemma 4.14.
Theorem 5.16. Assume that U is a trivial Guniverse. Let C be an Illmancol
lection of subgroups of G such that G 2 C. Then there is a cofibrantly generat*
*ed
proper simplicial tensor model structure on MR such that the weak equivalence
are Cunderlying equivalences and the cofibrations are retracts of relative Cc*
*ell
complexes.
Proof.This is a special case of Theorem 5.5 and Proposition 5.11.
We refer to this model structure as the Ccofree model structure on MR .
Remark 5.17. We require the universe to be trivial as part of the definition of
the Ccofree model structure. When {1} is in C there is no loss of generality in
making this assumption.
6.A digression: Gspectra for noncompact groups
In this section we consider an example of a model structure on orthogonal G
spectra where the homotopy theory is "pieced together" from the genuine homotopy
theory of the compact Lie subgroups of G. This example is inspired by conversat*
*ions
with Wolfgang L"uck. This section plays no role later in the paper.
The model structure we construct below in Proposition 6.5 is in many ways
opposite to the model structure (to be discussed) in Theorem 8.4: Compact Lie
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 29
subgroups versus compact Lie quotient groups, indspectra versus prospectra, p*
*ro
universes versus induniverses. The difficulties here lies in dealing with inv*
*erse
systems of universes for the finite subgroups of G.
Let G be a topological group, and let X be a trivial Guniverse. Let R be a
symmetric monoid in the category of orthogonal Gspectra indexed on X . Let M
denote the category of Rmodules indexed on X .
Definition 6.1. Let FG denote the family of compact Lie subgroups of G.
If G is a discrete group or a profinite group then FG is the family of finite
subgroups of G. The results in this section remains true if we replace FG with *
*any
collection of subfamilies such that whenever J < G and H 2 F0J, then H 2 F0G.
By Proposition 3.14 there is a cofibrantly generated model structure on M
such that the cofibrations are retracts of relative 1RFG Icell complexes and *
*the
weak equivalences are levelwise FG equivalences. We would like to stabilize M
with respect to Hrepresentations for all compact Lie subgroups H of G. An
Hrepresentation might not be a retract of a Grepresentation restricted to H
(there might not be any nontrivial Hrepresentations of this form).
Our approach is to localize M with respect to stable Hhomotopy groups de
fined using a complete Huniverse, one universe for each H in FG .
Definition 6.2. An FG universe consists of an Huniverse UH , for each H 2
FG , such that whenever H K, then UK is a subuniverse of UH K. For any
subgroups H K L the three resulting inclusions of universes are required to
be compatible.
We say that the FG universe, {UH }, is complete if UH is a complete Huniver*
*se
for each H 2 FG .
Lemma 6.3. There exists a complete FG universe.
Proof.Choose a complete Huniverse U0Hfor each H 2 FG . Let UH be defined to
be L
K H (U0KH)
where the sum is over all K 2 FG that contains H.
Let H be a compact Lie group. Then there is a stable model structure on
orthogonal Hspectra, indexed on a trivial Huniverse, that is Quillen equivale*
*nt
to the "genuine" model structure on orthogonal Hspectra indexed on a complete
Huniverse. This is proved in [35, V.1.7] (note that the condition V V0, used
there, is not necessary). The point of view of doing "genuine" stable equivaria*
*nt
homotopy theory in the category of spectra indexed on a trivial universe has be*
*en
advocated by Morten Brun.
Let H be a compact Lie group and let V and V0be collections of Hrepresentati*
*ons
containing the trivial Hrepresentations. Typically, V is the collection V(U) *
*of
all Hrepresentations that are isomorphic to some indexing representation in an
Huniverse U. There is a change of indexing functor
0 V
IVV0:JGVS ! JG S
defined in [35, V.1.2]. The functor IVV0is an equivalence of categories with IV*
*0Vas
the inverse functor. The functor IVV0is a strong symmetric tensor functor. Thes*
*e,
and other, claims are proved in [35, V.1.5].
30 H. FAUSK
Lemma 6.4. For each compact Lie subgroup H of G the functor
ssH*(IV(UH)V(X))
is a homology theory on M with the levelwise FG X model structure, that satis*
*fies
the colimit axiom.
See the discussion after Corollary 4.4. Note that any family of subgroups of H
is an UH Illman collection. (See Definition 4.2).
Proof.This follows since IVV0respects homotopy colimits and weak equivalences
since V(X ) V(U) [35, V.1.6].
We localize the stable FG  X model category with respect to the homology
theory given by L
h = H ssH*(IV(UH)V(X)),
where the sum is over all H 2 FG .
Proposition 6.5. Given an FG universe {UH }. Then there is a cofibrantly gen
erated proper stable model structure on M such that the cofibrations are retrac*
*ts of
relative 1RFG Icell complexes and the weak equivalences are the h*equivalenc*
*es.
If FG is an Illman collection of subgroups of G (see Def. 2.17), then the model
structure satisfies the pushoutproduct axiom.
This model structure is called the stable {UH }model structure on M.
Proof.See the proof of Theorem 5.5. The argument given there shows that the
model structure is proper. If FG is an Illman collection, then the model struct*
*ure
is tensorial by Lemma 5.10 and [35, III.3.11].
The cofibrant replacement of 1 S0 in this model structure (regardless of {UH*
* })
is given by 1 (EFG )+ , where EFG is an FG cell complex such that (EFG )H is
contractible whenever H 2 FG , and empty otherwise [35, IV.6].
Lemma 6.6. Assume G is a discrete group. If X is a Gcell complex, then
X ^ (EFG )+ is a cofibrant replacement of X.
Proof.Note that G=J+ ^ G=H+ is an FG cell complex, whenever H 2 FG and
J is an arbitrary subgroup of G. The collapse map (EFG )+ ! S0 induces an
FG equivalence X ^ (EFG )+ ! X
Lemma 6.7. If G has no compact Lie subgroups besides {1} (e.g. torsionfree dis
crete groups), then the stable {UH }model structure M is the stable model stru*
*cture
with underlying weak equivalences.
Lemma 6.8. If G is a compact Lie group, then the stable {UH }model structure
on M is Quillen equivalent to the {all}  UG model structure on M.
Let J be a subgroup of G. Let R be a monoid in the category of orthogonal
Gspectra. Let M denote the category of Rmodules in the category of orthogonal
Gspectra indexed on X , and let M0 denote the category of RJmodules in the
category of orthogonal Jspectra indexed on X J. Let {UH } be an FG universe,
and set {UH }H2FJ be the FJuniverse.
Note that the condition in the next Lemma is trivially satisfied if G is a di*
*screte
group.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 31
Lemma 6.9. Assume that G=K+ has the structure of a J  FJcell complex for
any K 2 FG . Give M the stable FG  {UH }model structure, and give M0 the
FJ  {UH }H2FJ model structure. Then the functor
FJ(G+ , ): MJ ! MG
is right Quillen adjoint to the restriction functor
MG ! MJ.
Proof.The restriction functor from Gspectra to Jspectra respects weak equiva
lences by the definition of weak equivalences. Since G=K is a J  FJcell compl*
*ex
for all K 2 FG , by our assumption, the relative G  FG cell complexes are also
relative J  FJcell complexes.
Lemma 6.10. Assume G is a discrete group. Give M the stable FG {UH }model
structure, and give M0 the stable FJ  {UH }H2FJ model structure. Then the
functor
G+ ^J  :MJ ! MG
is left Quillen adjoint to the restriction functor
MG ! MJ.
Proof.Since G+ ^J J=K+ ~=G=K+ , and the functor G+ ^J  respects change of
universe functors and colimits, it follows that G+ ^J  respects cofibrations.
Let f :X ! Y be a J FJequivalence. We observe that G+ ^JX is isomorphic
to
W
HgJ2H\G=J H+ ^gJg1\H gX
as an orthogonal Hspectrum. Hence the map G+ ^Jf is an FG weak equivalence
if each H+ ^gJg1\H g(f) is a Hequivalence for H 2 FG . This follows from [35,
V.1.7,V.2.3] since g(f) is a Kequivalence for every K gJg1 \ H, because
K 2 FG and K gJg1 implies that K 2 FgJg1.
Lemma 6.11. Assume G is a discrete group. Let X be in M0and let Y be in M.
Then
[G+ ^J X, Y ]G ~=[X, (Y J)]J,
where the first homgroup is in the homotopy category of the {UH }model struc
ture on Rmodules, and the second homgroup is in the homotopy category of the
{UH }H2FH model structure on RHmodules.
In particular, if X and Y are Gspectra and H 2 FG , then
[G=H+ ^ X, Y ]G ~=[X, Y ]H .
Remark 6.12. A better understanding of the fibrations, or at least the fibrant
objects would be useful. They are completely understood when G is a compact Lie
group [35, III.4.7,4.12]. Calculations in the stable {UH }homotopy theory redu*
*ces
to calculations in the stable homotopy categories for the compact Lie subgroups*
* of
G (via a spectral sequence). This follows from Lemma 6.10 using a cell filtrati*
*on
of a cofibrant replacement of the source by a FG cell complex.
32 H. FAUSK
7. Postnikov tmodel structures
We modify the construction of the WCmodel structure on MR by considering
the n  Wequivalences, for all n, instead of Wequivalences. This is used when
we give model structures to the category of prospectra, pro MR , in Section 8*
*.2.
The homotopy category of a stable model category is a triangulated category
[26, 7.1]. We consider tstructures on this triangulated category together with
a lift of the tstructure to the model category itself. The relationship betwe*
*en
nequivalences and tstructures is given below in Definition 7.4 and Proposition
7.10.
7.1. Preliminaries on tmodel categories. We recall the terminology of a t
structure [3, 1.3.1] and of a tmodel structure [20, 4.1].
Definition 7.1. A homologically graded tstructure on a triangulated category
D, with shift functor , consists of two full subcategories D 0 and D 0 of D,
subjected to the following three axioms:
(1) D 0 is closed under , and D 0 is closed under 1;
(2) for every object X in D, there is a distinguished triangle
X0! X ! X00! X0
such that X02 D 0 and X002 1D 0; and
(3) D(X, Y ) = 0, whenever X 2 D 0 and Y 2 1D 0.
For convenience we also assume that D 0 and D 0 are closed under isomorphisms
in D.
Definition 7.2. Let D n = nD 0, and let D n = nD 0.
Remark 7.3. A homologically graded tstructure (D 0, D 0) corresponds to a co
homologically graded tstructure (D 0, D 0) as follows: D n = D n and D n =
D n .
Definition 7.4. The class of nequivalences in D, denoted Wn, consists of all
maps f :X ! Y such that there is a triangle
F ! X f!Y ! F
with F 2 D n . The class of conequivalences in D, denoted coWn, consists
of all maps f such that there is triangle
X f!Y ! C ! X
with C 2 D n .
If D is the homotopy category of a stable model category K, then a map f in K*
* is
called a (co)nequivalence if the corresponding map f in the homotopy category,
D, is a (co)nequivalence. We use the same symbols Wn and coWn for the classes
of nequivalences and conequivalence in K and D, respectively.
Definition 7.5. A tmodel category is a proper simplicial stable model category
K equipped with a tstructure on its homotopy category together with a functori*
*al
factorization of maps in K as an nequivalence followed by a conequivalence in
K.
Tmodel categories are discussed in detail in [20]. They give rise to interes*
*ting
model structures on procategories.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 33
7.2. The dPostnikov tmodel structure on MR . We construct tmodel struc
tures on MR making use of Postnikov sections. In Section 8 we use this tmodel
structure to produce model structures on the category of prospectra. We allow
Postnikov sections where the cutoff degree of ssH*depends on H.
Construction 7.6. Assume that D is the homotopy category of a proper simplicial
stable model category M. Let D be the homotopy category of M. Let D 0 be
a strictly full subcategory of D that is closed under . Define D n to be nD 0.
Define Wn as in Definition 7.4, and lift Wn to M. Let C denote the class of
cofibrations in M, and define Cn = Wn \ C and Fn = injCn. Let D n+1 be the
full subcategory of D with objects isomorphic to hofib(g) for all g 2 Fn. If th*
*ere
is a functorial factorization of any map in M as a map in Cn followed by a map
in Fn, then D 0, D 0 is a tstructure on D. Hence the model category M, the
factorization, and the tstructure on D is a tmodel structure on M [20, 4.12].
Let C and W be collections of subgroups of G such that CW C. Let R be a
ring, and let D be the homotopy category of WCMR .
Definition 7.7. A class function on W is a function d: W ! Z [ {1, 1}
such that d(H) = d(gHg1), for all H 2 W and g 2 G.
Definition 7.8. Let d be a class function. Define a full subcategory of D by
Dd 0= {X  Ui(X) = 0 fori < d(U), U 2 W}.
Let Dd 1be the full subcategory of D given by Construction 7.6.
If H is in the closure of a sequence Ha and ssHan(X) = 0, then ssHn(X) = 0,
for a Gspace X. Hence it suffices to consider continuous class functions, whe*
*re
Z [ {1, 1} has the topology given by letting the open sets be {n N}, for
N 2 Z [ {1, 1}, and W has the Hausdorff topology.
The next result is needed to get a tmodel structure on MR .
Lemma 7.9. Any map in MR factors functorially as a map in Cn followed by a
map in Fn. Moreover, there is a canonical map from the nth factorization to the
(n  1)th factorization.
Proof.The proof is similar to the proof of Theorem 5.5. See also [16, Appendix].
Lemma 7.10. Let d be a class function on W. The W  Cmodel structure on
MR , the two classes Dd 0and Dd 0, together with the factorization in Lemma 7.9
is a tmodel structure.
Proof.This follows from Theorem 5.5 and [20, 4.12].
This tmodel structure is called the dPostnikov tmodel structure on
WCMR . We call the 0Postnikov tmodel category simply the Postnikov tmodel
category.
A map f of spectra is an nequivalence with respect to the dPostnikov t
structure if and only if Um(f) is an isomorphism for m < d(U) + n and Ud(U)+n*
*(f)
is surjective for all U 2 W.
34 H. FAUSK
7.3. An example: Greenlees connective Ktheory. To show that there is
some merit to the generality of dPostnikov tstructures, we recover Greenlees *
*equi
variant connective Ktheory as the dconnective cover of equivariant Ktheory
for a suitable class function d. Let G be a compact Lie group, and let W = C be
the class of all closed subgroups of G. Note that {1} is an open closed point i*
*n C.
(In fact, any family gives an open subset of C by Montgomery and Zippin's theor*
*em
[38].) Let Pn denote the nth Postnikov section functor, and let Cn denote the *
*nth
connective cover functor.
Lemma 7.11. Let G be a compact Lie group. Let d be the class function such that
d({1}) = 0 and d(H) = 1 for all H 6= {1}. Then
X n = F (EG+ , PnX)
is a functorial truncation functor for the dPostnikov tmodel structure on WMR*
* .
The nth dconnective cover is given by the homotopy pullback of the left most
square in the diagram
(7.12) X n _____________//X_________//F (EG+ , Pn+1X)
  
  
fflffl fflffl 
F (EG+ , CnX)____//_F (EG+ ,_X)__//F (EG+ , Pn+1X).
In particular, (KG ) 0 is Greenlees' equivariant connective Ktheory [22, 3.1].
Proof.Axiom 1 of a tstructure is satisfied since
F (EG+ , Pn+1X) ~=F (EG+ , Pn+1X).
We combine the verification of axioms 2 and 3 of a tstructure. Let X n denote
the homotopy fiber of the natural transformation X ! F (EG+ , Pn+1X). Since
X ! F (EG+ , X) is a nonequivariant equivalence we conclude, using Diagram
7.12, that X n and CnX are nonequivariant equivalent. Hence X n 2 D n for all
X 2 D. If Y 2 D n and X 2 D, then
D(Y, F (EG+ , Pn+1X)) = 0
since Y ^ EG+ is in CnD.
This example can also be extended to arbitrary compact Hausdorff groups [18].
7.4. Postnikov sections. Suppose d is a constant function and R has trivial
Whomotopy groups in negative degrees. Then there is a useful description of
the full subcategory D 0 of the homotopy category D of WCMR .
Definition 7.13. We say that a spectrum R is Wconnective if Un(R) = 0 for
all n < 0 and all U 2 W.
In other words R is Wconnective if R 2 D 0 for the Postnikov tstructure on
WCMR .
Proposition 7.14. Let R be a Wconnective ring. Then there is a tstructure on
the homotopy category D of WCMR defined by the two full subcategories of D:
D 0 = {X  Ui(X) = 0 wheneveri < 0, U 2 W}
and
D 0 = {X  Ui(X) = 0 wheneveri > 0, U 2 W}.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 35
Proof.Recall that the full subcategory D 1 of D has objects Y such that D(X, Y *
*) =
0 for all X 2 D 0 [3, 1.3.4]. Proposition 4.25 gives that G=H+ ^ Sn 2 D 0, for *
*all
n 0 and all H 2 C. This gives that
D 1 {Y  Ui(Y ) = 0 wheneverU 2 W, i 0}.
The converse inclusion is proved in two steps. We first prove it when W
C. Assume that X 2 D 0 and that Ui(Y ) = 0, whenever U 2 W and i 0.
By Lemma 4.23 there is a Ccell complex approximation X0 ! X such that X0
is a cell complex built from cells in nonnegative dimensions, and X0 ! X is a
Cisomorphism in nonnegative degrees, hence a Wequivalence. So we get that
D(X, Y ) = 0. Since this is true for all X 2 D 0 we conclude that Y 2 D 1.
We now consider the general case. Assume that X 2 D 0 and that Ui(Y ) = 0,
whenever U 2 W and i 0. By the first part of the proof there are Ctruncations
k :X0 ! X and l :Y ! Y 0such that: (1) ssHi(X0) = 0, for all H 2 C and
i < 0, ssHi(k) is an isomorphism, for all H 2 C and i 0, (2) ssHi(Y 0) = 0, f*
*or all
H 2 C and i 0, and ssHi(l) is an isomorphism, for all H 2 C and i < 0. Our
assumptions on X and Y give that X0 ! X and Y ! Y 0are Wequivalences.
Hence D(X, Y ) = D(X0, Y 0) vanish. Since X 2 D 0 was arbitrary we conclude
that Y 2 D 1. The result follows.
Hence a map g is a conequivalence in the (0)Postnikov tstructure if and
only if Wm(g) is an isomorphism for m > n and Un(g) is injective for U 2 W.
When the universe is trivial we can give a similar description of the tmodel
structure for more general functions d. We say that a class function d: W !
Z [ {1, 1} is increasing if d(H) d(K) whenever H K.
Proposition 7.15. Assume the Guniverse U is trivial, and let R be a Wconnecti*
*ve
ring. Let d be an increasing class function. Then there is a tstructure on the*
* ho
motopy category D of WCMR defined by the two full subcategories of D:
Dd 0= {X  Ui(X) = 0 wheneveri < d(U), U 2 W}
and
Dd 0= {X  Ui(X) = 0 wheneveri > d(U), U 2 W}.
Proof.This follows from 4.27, the proof of Proposition 7.14, and [20, 4.12].
Definition 7.16. Let X be a spectrum in MR . The nth Postnikov section of X
is a spectrum PnX together with a map pnX :X ! PnX such that Um(PnX) = 0,
for m > n and all U 2 W, and m (pnX): Um(X) ! Um(PnX) is an isomorphism,
for all m n and all U 2 W. A Postnikov system of X consists of Postnikov
factorization pn :X ! PnX, for every n 2 Z, together with maps rnX :PnX !
Pn1X, for all n 2 Z, such that rnX O pnX = pn1X.
Dually, one defines the nth connected cover CnX ! X of X. The nth
connected cover satisfies Uk(CnX) = 0, for k n, and Uk(CnX) ! Uk(X) is an
isomorphism, for k > n.
Definition 7.17. A functorial Postnikov system on MR consists of functors
Pn, for each n 2 Z, and natural transformation pn :1 ! Pn and, rn :Pn ! Pn1
such that pn(X) and rn(X), for n 2 Z, is a Postnikov system for any spectrum X.
Proposition 7.18. Let R be a Wconnective ring. Then the category WCMR
has a functorial Postnikov system.
36 H. FAUSK
Proof.This follows from Lemma 7.9.
Remark 7.19. Classically, one also requires that the maps rnX are fibrations for
every n and X. We can construct a functorial Postnikov tower with this property,
if we restricted ourself to the full subcategory D n for some n [20, Sect.7].
7.5. Coefficient systems. In this subsection we describe the EilenbergMac Lane
objects in the Postnikov tstructure. Let C be a UIllmancollection. Let W be a
collection such that WC C. Let R be a Wconnective ring spectrum.
Definition 7.20. The heart of a tstructure (D 0, D 0) on a triangulated catego*
*ry
D is the full subcategory D 0 \ D 0 of D consisting of objects that are isomorp*
*hic
to object both in D 0 and in D 0.
The heart of a tstructure is an abelian category [3, 1.3.6].
Definition 7.21. An Rmodule X is said to be an EilenbergMac Lane spec
trum if Un(X) = 0, for all n 6= 0 and all U 2 W.
Lemma 7.22. Let C and W be collections of subgroup of G such that CW C,
and let R be a Wconnective ring spectrum. If d: W ! Z is the 0function, then
the heart of the homotopy category of WCMR is the full subcategory consisting of
the EilenbergMac Lane spectra.
Proof.This follows from Proposition 7.14.
We give a more algebraic description of the heart in terms of coefficient sys*
*tems
when W C. Let D denote the homotopy category of WCMR .
Definition 7.23. The orbit category, O, is the full subcategory of D with objec*
*ts
1RG=H+ , for H 2 W.
This definition makes sense since W is contained in C. The orbit category de
pends on G, W, C, and the Guniverse U.
Definition 7.24. A W Rcoefficient system is a contravariant additive func
tor from Oop to the category of abelian groups.
Denote the category of W  Rcoefficient systems by G. This is an abelian
category. An object Y in D naturally represents a coefficient system given by
1RG=H+ 7! D( 1RG=H+ , Y ).
We make use of Lemma 4.8 in the next definition.
Definition 7.25. Let X be an Rmodule spectrum. The nth homotopy coefficient
system of X, ssWn(X), is the coefficient system naturally represented by X ^ RR*
*nS0
when n is positive, and by X ^ 1RSn when n is negative.
Lemma 7.26. There is a natural isomorphism
G(ssW0( 1RG=H+ ), M) ~=M(G=H)
for any H 2 W.
Proof.This is an immediate consequence of the Yoneda Lemma.
Proposition 7.27. Let R be a Wconnected ring spectrum. The functor ssW0 in
duces a natural equivalence from the full subcategory of EilenbergMac Lane spe*
*ctra
in the homotopy category of WCMR , to the category of W Rcoefficient systems.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 37
Proof.We need to show that for every coefficient system M, there is a spectrum
HM such that ssW0(HM) is isomorphic to M as a coefficient system, and further
more, that ssW0 induces an isomorphism D(HM, HN) ! GW (M, N) of abelian
groups.
We construct a functor, H, from G to the homotopy category of spectra. The
natural isomorphism in Lemma 7.26 gives a surjective map of coefficient systems
L L
f(M): H2W M(G=H+) ss0(G=H+ ) ! M.
This construction is natural in M. Let CM be the kernel of f(M) and repeat the
construction with CM in place of M. We get an exact sequence
(7.28)L L L L
K2W CM (G=K+)ssW0(G=K+ ) ! H2W M(G=H+)ssW0(G=H+ ) ! M ! 0.
This sequence is natural in M. We have that H2W M(G=H+) ssW0(G=H+ ) is
naturally isomorphic to
W W
ssW0( H2W M(G=H+) G=H+ )
and W W
GW (ssW0(G=K+ ), ssW0( H2W M(G=H+) G=H+ ))
is naturally isomorphic to
W W
D(G=K+ , H2W M(G=H+) G=H+ ).
Hence there is a map
W W W W
h(M): K2W CM (G=K+)G=K+ ! H2W M(G=H+) G=H+ ,
unique up to homotopy, so ssW0(h(M)) is isomorphic to the leftmost map in the
exact sequence 7.28. Proposition 4.25 says that ssWn(G=K+ ) = 0, for all n <
0 and K 2 W. So ssn(hocofib(h(M))) = 0, for n < 0, and there is a natural
isomorphism ssW0(hocofib(h(M))) ~=M. Now define HM to be the zeroth Postnikov
section, P0hocofib(h(M)), of the homotopy cofiber of h(M). We get that HM is
an EilenbergMac Lane spectrum and there is a natural isomorphism
ssW0(HM) ~=M.
Conversely, let X be an EilenbergMac Lane spectrum. Then there is a natural
equivalence
HssW0(X) ! X
in D. This proves the first part.
The map hocofib(h(M)) ! HM induces an isomorphism
[HM, HN] ! [hocofib(h(M)), HN].
We then get an exact sequence
W W W W
0 ! [HM, HN] ! [ H2W M(G=H+) G=H+ , HN] ! [ K2W C(G=K+)G=K+ , HN],
where the rightmost map is induced by h(M) and the leftmost map is injective.
Applying ssW0 gives an isomorphism between the last map and the map
L L
G( H2W,M(G=H+) ssW0(G=H+ ), N) ! G( K2W,C(G=K+) ssW0(G=K+ ), N).
The kernel of this map is G(M, N), so ssW0 induces an isomorphism
[HM, HN] ! G(M, N)
38 H. FAUSK
of abelian groups.
Remark 7.29. When d is not a constant class function, then the homotopy groups
of the objects in the heart need not be concentrated in one degree. For exam
ple the heart of the tstructure in Lemma 7.11 consists of spectra of the form
F (EG+ , HM), where M is an EilenbergMac Lane spectrum. The heart of the
Postnikov tstructure on D is not well understood for general functions d and n*
*on
connective ring spectra R.
7.6. Continuous Gmodules. When W 6 C it is harder to describe the full
subcategory of EilenbergMac Lane spectra as a category of coefficient systems.*
* We
give_a_description_of the heart of the Postnikov tstructure on the homotopy ca*
*tegory
of fnt(G)free model structure on MR when G is a compact Hausdorff group.
Let R0 denote the (continuous) Gring colimUssU0(R). We have that {1}0(X) ~=
colimUssU0(X) is a continuous R0  Gmodule.
Proposition 7.30. The heart of D is equivalent to the category of continuous
R0  Gmodules and continuous Ghomomorphisms between them.
Proof.Let M be a continuous R0  Gmodule. For any m 2 M let st(m) denote
the stabilizer, {g 2 G  gm = m}, of m. We get a canonical surjective map
L
h: R0[G=st(m)] ! M
where the sum is over all elements m in M. The map R[G=st(m)] ! M, corre
sponding to the summand m, is given by sending the element (r, g) to r.gm. This*
* is
a Gmap since g0r . g0gm = g0(rgm), for g02 G. Repeating this construction with
the kernel of h gives a canonical right exact sequence of continuous R0Gmodul*
*es
L f L
(7.31) R0[G=U0] ! R0[G=U] ! M ! 0.
We want to realize this sequence on spectra level. We have that
{1}0(R ^ G=U+ ) ~=R0[G=U]
______ {1}
as R0  Gmodules, for all U 2 fnt(G). The map f is realized as 0 applied
to a map h(M): _ R ^ G=U0+! _R ^ G=U+ . Let Z be the homotopy cofiber of
h(M). The right exact sequence in 7.31 is naturally isomorphic to {1}0applied *
*to
the sequence W W
R ^ G=U0+! R ^ G=U+ ! Z.
This follows from Proposition 4.25. We define HM to be the 0th Postnikov sectio*
*n,
P0(Z), of Z. Proposition 4.25 gives that {1}n(HM) = 0 when n 6= 0, and there
is a natural isomorphism {1}0(HM) ~=M of continuous R0  Gmodules, for any
continuous R0 Gmodule M, and there is a natural equivalence H {1}0(X) ! X
in D, for an object X in the heart. It remains to show that {1}0is a full and *
*faithful
functor. The same argument as in the proof of Proposition 7.27 applies.
8. ProGspectra
In this section we use the W CPostnikov tmodel structure on MR , discussed
in Section 7, to give a model structure on the procategory, pro MR . For term*
*i
nology and general properties of procategories see for example [20, 31]. We re*
*call
the following.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 39
Definition 8.1. Let M be a collection of maps in C. A levelwise map g = {gs}s2S
in pro C is a levelwise Mmap if each gs belongs to M. A promap f is an
essentially levelwise Mmap if f is isomorphic, in the arrow category of proC,
to a levelwise Mmap. A map in pro C is a special Mmap if it is isomorphic
to a cofinite cofiltered levelwise map f = {fs}s2S with the property that for e*
*ach
s 2 S, the map
Msf :Xs ! limt m. Hence the map from the second expression to 10.19 is an equivalence.
Since (EG=N+ )n only has cells in dimension less than or equal to n we get th*
*at
F ( 1 (EG=N+ )(n), Y ) ! F ( 1 (EG=N+ )(n), Pm Y )
is an equivalence when n < m. Hence the map from the first expression to 10.19 *
*is
an equivalence.
Hence, the spectrum associated to our definition of the homotopy fixed point
prospectrum agrees with Davis' definition when G is a profinite group with fin*
*ite
virtual cohomological dimension.
Corollary 10.20. If Y is a strict fibrant (commutative) Ralgebra in pro MR ,
then Y hGK, for all K, are also (commutative) Ralgebras in pro MR .
Proof.By Proposition 10.18 we get that Y hGK is equivalent to
{hocolimU(hocolimNF ( 1 (EG=N+ )(n), Yb))UK }b,m.
The result follows since the procategory is cocomplete by [20, 11.4], directed*
* colim
its of algebras are created in the underlying category of modules, and fixed po*
*ints
preserves algebras.
EQUIVARIANT HOMOTOPY THEORY FOR PROSPECTRA 55
Appendix A. Compact Hausdorff Groups
In this appendix we recall some well known properties of compact Lie groups.
We show that the relationship between compact Lie groups and compact Hausdorff
groups are analogue to the relationship between finite groups and profinite gro*
*ups.
We give a point set topological remark. Since we work in the category of
weak Hausdorff spaces closed subgroups of compact spaces are again compact.
We first note that if G is a compact Hausdorff group, then the finite dimension*
*al
Grepresentations are all obtained from G=Nrepresentations via a suitable quo
tient map G ! G=N where G=N is a compact Lie group quotient of G.
Lemma A.1. Let V be a finite dimensional Grepresentation. Then the Gaction
on V factors through some compact Lie group quotient G=N of G.
Proof.A Grepresentation V is a group homomorphism
ae: G ! GL (V ).
The action factor through the image ae(G). Since G is a compact group ae(G),
with the subspace topology from GL (V ), is a closed subgroup of the Lie group
GL (V ). Hence ae(G) is itself a Lie group. Again, since G is compact Hausdorff*
*, the
subspace topology on ae(G) agrees with the quotient topology from ae. Hence we
have a homeomorphism G= kerae ~=ae(G); and G= kerae is a compact Lie group.
Recall from Example 2.7 that Lie(G) denotes the collection of closed normal
subgroups, N, of G such that G=N is a compact Lie group. We consider the inverse
system of quotients G=N such that G=N is a compact Lie group. If G=N and G=K
are compact Lie groups, then G=N \ K is again a compact Lie group, since it is
a closed subgroup of G=N x G=K. Hence the inverse system is a filtered inverse
system.
In the next theorem it is essential that we work in the category of weak Haus*
*dorff
compactly generated topological spaces.
Proposition A.2. Let X be a topological space with a (not necessarily continuou*
*s)
Gaction. Then the Gaction on X is continuous_if_and only if the action by G
on X=N is continuous for all subgroups N 2 Lie(G)and the canonical map
ae: X ! limNX=N,
where the limit is over all N 2 Lie(G), is a homeomorphism.
Proof.Assume that ae is a homeomorphism. Then the Gaction on X is continuous
since the Gaction on limNX=N is continuous.
We now assume that the Gaction on X is continuous. We first show that
ae: X ! limNX=N
is a bijection. The PeterWeyl theorem for compact Hausdorff groups implies that
there are enough finite dimensional real Grepresentations to distinguish any t*
*wo
given elements in G [1, 3.39]. Hence \N N is {1}, and ae is injective. Now let *
*{NxN }
be an element in limNX=N. Since G is a compact group and since the Gaction
on X is continuous we get, for every N 2 Lie(G), that NxN is a compact subset of
X. We get that GxN = GxV for all N and V in Lie(G). We denote this compact
set K. Since \N N = 1 and NxN \ V xV N \ V xN\V , we conclude that the
intersection of the closed sets NxN , for N 2 Lie(G), is a point. Call this poi*
*nt x.
We then have that ae(x) = {NxN }. So ae is surjective.
56 H. FAUSK
We need to show that ae is a closed map. This amounts to showing that for any
closed set A of X, and for any N 2 Lie(G) we have that N . A is a closed subset
of X. When A is a compact (hence closed) subset of X this follows since N . A
is the image of N x A under the continuous group action on X. Since we use the
compactly generated topology the subset N . A of X is closed if for all compact
subsets K of X the subset (N . A) \ K is closed in X. This is true since
(N . A) \ K = (N . (A \ (N . K))) \ K
and N . K is a compact subset of X. Hence ae is a homeomorphism.
Corollary A.3. Any compact Hausdorff group G is an inverse limit of compact
Lie groups.
Proof.This follows from Theorem A.2 by letting X be G.
Corollary A.4. The category of GT is a retract of the category pro GLie(G)T .
Proof.A Gspace X is sent to the pro GLie(G)space {X=N}. The retract map
is given by taking the inverse limit. By Theorem A.2 the composite is isomorphic
to the identity map on GT . Let X and Y be two Gspaces. We have that
GT (X, Y ) ! limNcolimN GT (X=N, Y=V )
is a bijection (but not necessarily a homeomorphism).
Remark A.5. Let G be a profinite group. We observe that in the category of sets,
X is a continuous Gset if and only if
colimN XN ! X
is a bijection. On the other hand, in the category of compactly generated space*
*s,
X is a continuous Gspace if and only if X ! limNX=N is a continuous Gspace.
It is worth mentioning that the category of procompact Lie groups is equival*
*ent
to the category of compact Hausdorff groups. This follows since a closed subgro*
*up
of a compact Lie group is again a compact Lie group. Actually, the categories
are equivalent as topological categories (both homspaces are compact Hausdorff
spaces).
Groups which are inverse limits of Lie groups have been studied recently. See
for example [24].
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Email address: fausk@math.uio.no
Department of Mathematics, University of Oslo, 1053 Blindern, 0316 Oslo, Norw*
*ay