THE DUALIZING SPECTRUM
OF A
TOPOLOGICAL GROUP
JOHN R. KLEIN
Abstract. To a topological group G, we assign a naive G-spectrum
DG , called the dualizing spectrum of G. When the classifying space
BG is finitely dominated, we show that DG detects Poincare du-
ality in the sense that BG is a Poincare duality space if and only
if DG is a homotopy finite spectrum. Secondly, we show that the
dualizing spectrum behaves multiplicatively on certain topologi-
cal group extensions. In proving these results we introduce a new
tool: a norm map which is defined for any G and for any naive
G-spectrum E.
Applications of the dualizing spectrum come in two flavors: (i)
applications in the theory of Poincare duality spaces, and (ii) ap-
plications in the theory of group cohomology.
On the Poincare duality space side, we derive a homotopy the-
oretic solution to a problem posed by Wall which says that in
a fibration sequence of finitely dominated spaces, the total space
satisfies Poincare duality if and only if the base and fiber do. The
dualizing spectrum can also be used to give an entirely homotopy
theoretic construction of the Spivak fibration of a finitely dom-
inated Poincare duality space. We also include a new proof of
Browder's theorem that every finite H-space satisfies Poincare du-
ality.
In connection with group cohomology, we show how to define a
variant of Farrell-Tate cohomology for any topological or discrete
group G, with coefficients in any naive equivariant cohomology
theory E. When E is connective, and when G admits a subgroup
H of finite index such that BH is finitely dominated, we show
that this cohomology coincides with the ordinary cohomology of
G with coefficients in E in degrees greater than the cohomological
dimension of H.
In an appendix, we identify the homotopy type of DG for certain
kinds of groups. The class includes all compact Lie groups, torsion
free arithmetic groups and Bieri-Eckmann duality groups.
____________
Date: August 17, 2000.
1991 Mathematics Subject Classification. Primary: 55P91, 55N91, 55P42,
57P10. Secondary: 55P25, 20J05, 18G15.
1
2 JOHN R. KLEIN
1. Introduction
In this paper the symbol G will denote either the realization of a
simplicial group or a Lie group. Let
S0[G]
(the "group ring of G over the sphere spectrum") denote the suspension
spectrum of G+ , i.e., the spectrum whose j-th space is Q(Sj ^ (G+ )),
where Q is the stable homotopy functor (here and elsewhere, G+ de-
notes the union of G with a disjoint basepoint).
Let Gx G act on G by the rule (g; h) * x = gxh-1. This induces a
left action of Gx G on S0[G].
Definition. The dualizing spectrum of G is the homotopy fixed point
spectrum of the subgroup G = Gx 1 Gx G acting on S0[G]:
DG = S0[G]hG := F (EG+ ; S0[G])Gx 1:
This is a G-spectrum, whose action is given by restriction to the sub-
group 1x G Gx G.
We emphasize that this work will only employ the naive kind of
equivariant spectra (that is, the group doesn't act on the suspension
coordinates).
Motivation. In making this definition, we were prompted by a sim-
ilar construction arising in the theory of group cohomology. Given
a discrete group whose classifying space is finitely dominated, one
considers
D := hom D(Z[])(Z; Z[])
where hom is taken internally within the derived category of (left) Z[]-
modules (the homology of this complex is, of course, Ext *Z[](Z; Z[])).
One calls D a dualizing module if it is isomorphic in the derived cate-
gory to a complex which is non-trivial in a single degree - n and which
in that degree is torsion free as an abelian group (compare [Br4 , Ch.
VIII Th. 10.1]). If D is a dualizing module, then B satisfies a version
of Poincare duality in which the fundamental class lives in Hn(BG; D ).
Such groups are called Bieri-Eckmann duality groups.
By analogy, our dualizing spectrum is given by replacing the discrete
by the (possibly) continuous G, and the integers Z by the sphere S0:
DG = hom D(S0[G])(S0; S0[G]) ;
where hom is now taken internally in the derived category of naive
G-spectra.
THE DUALIZING SPECTRUM 3
An important property of the dualizing spectrum is its ability to
detect Poincare duality in the classifying space BG:
Theorem A. Assume that BG is finitely dominated. Then the follow-
ing are equivalent:
1. BG is a Poincare duality space,
2. DG has the (unequivariant) weak homotopy type of a sphere,
3. DG is unequivariantly homotopy finite.
Furthermore, in (1) BG has formal dimension n, if and only if in (2)
DG is a sphere of dimension - n, if and only if in (3) DG has non-trivial
spectrum homology in degree - n.
The implication 1 , 2 is also due to Bill Dwyer (independently).
Theorem A will be proved in x5. In x10 (cf. x10 Ex. 1 and 10.5) we
refine Theorem A by identifying the equivariant weak homotopy type
of DG for those groups having finitely dominated classifying space.
Remark 1.1. Any connected based space X can be regarded up to ho-
motopy as BG for a suitable topological group G (take G to be a
topological group model for the based loop space of X). Consequently,
Theorem A characterizes the class of Poincare duality spaces.
Furthermore, if one regards the Borel construction
EGx GDG ! BG
as a "family of spectra" parametrized by points of BG, then Theorem
A shows that this fibration stably spherical precisely when BG is a
Poincare space. We show in Corollary 5.1 that the above is just the
Spivak normal fibration of BG. We therefore have a purely homotopy
theoretic construction the Spivak normal fibration.
Another feature of the dualizing spectrum is that it behaves multi-
plicatively with respect to certain kinds of extensions. Suppose that
1 ! H ! G ! Q ! 1
is an extension.
Theorem B. Assume either that
o the classifying spaces BH; BG and BQ are finitely dominated, or
that
o BH is a finitely dominated Poincare space.
Then there is a weak equivalence of spectra
DG ' DH ^ DQ :
4 JOHN R. KLEIN
Actually, Theorem B can be made equivariant. Call a map of G-
spectra an equivariant weak equivalence if it induces an isomorphism
on homotopy groups. More generally, two G-spectra X and Y are
equivariantly weak equivalent, written X 'G Y , if there exists a finite
zig-zag of such morphisms starting with X and ending at Y .
Using the fact that H is normal in G, it is possible to replace DH
by a G-spectrum D0H up to canonical weak equivalence of H-spectra
(cf. 2.6). Also, since Q acts on DQ as well, G acts on DQ by restriction
using the homomorphism G ! Q. Thus, we may give D0H^ DQ the
associated diagonal G-action.
Addendum C. With respect to the hypotheses of Theorem B, there is
a weak equivalence of G-spectra DG 'G D0H^ DQ .
The norm map. An important tool of this paper is the existence of
a norm map relating `invariants = group homology' to `coinvariants =
group cohomology.'
Theorem D. For any G-spectrum E, there is a (weak) map
DG ^hG E ! EhG
(natural in E) called the "norm map" which is a weak equivalence if
one of the following holds:
o G is arbitrary and E is G-finitely dominated in the sense that it
is a retract up to homotopy of a spectrum built up from a point by
attaching a finite number of free G-cells, or
o BG is finitely dominated and E is arbitrary, or
o G is a compact Lie group and E is an induced spectrum (in the
sense that E has the equivariant weak homotopy type of a spectrum
of the form W ^ G+ ).
Conversely, assume that ss0(G) is finitely presented, and that the norm
map is a weak equivalence for all G-spectra. Then BG is finitely dom-
inated.
Remarks 1.2. (1). The domain of the norm map is the homotopy orbit
spectrum of G acting diagonally on DG ^E. The codomain of the norm
map is the homotopy fixed point spectrum of G acting on E.
(2). To see how the norm map connects with Poincare duality, con-
sider the case when E = HM is the Eilenberg-Mac Lane spectrum on
a ss0(G)-module M. If BG is finitely dominated, and if DG is unequiv-
ariantly a sphere of fixed dimension - n, say, then applying homotopy
groups to the norm map gives an isomorphism
Hn-*(G; DG M) ~=H*(G; M) ;
THE DUALIZING SPECTRUM 5
where DG denotes ss- n(DG ). But this means that BG is a Poincare
duality space. (This gives the 2 ) 1 implication of Theorem A.)
(3). Assume BG is finitely dominated. Then the fact that the norm
map is a weak equivalence shows that taking homotopy fixed points
with respect to G commutes with homotopy colimits of G-spectra.
(4). The proof of Theorem D appears in x3, except in the instance when
G is a compact Lie group and E is induced. The proof of the latter
appears separately in 10.2. It is a consequence of the identification in
the compact Lie case
S0[G] 'Gx G F (G+ ; SAd G) ;
where the right side is a function spectrum of maps from G+ to SAd G=
the suspension spectrum of the one point compactification of the ad-
joint representation of G.
It should be true, although I haven't verified it, that our norm map
in the compact Lie case coincides with the norm map of Adem, Cohen
and Dwyer [A-C-D ] and Greenlees and May [G-M ].
The homotopy type of DG . In the appendix, we identify the weak
homotopy type of the dualizing spectrum for various kinds of groups
(sometimes it will be possible to identify the weak equivariant homo-
topy type). Here is the list of such groups (in the following, always
refers to a discrete group, while G can be either discrete or continuous;
a y indicates that the weak equivariant homotopy type is identified):
1. BG is a connected finitely dominated Poincare duality spacey
2. G is a compact Lie groupy
3. G is a finitely dominated topological group
4. is a discrete cocompact subgroup of a connected Lie group
5. = Z*g is the free group on g generatorsy
6. = Pn is the pure braid group on n-strings
7. = Z2 * Z2 is the infinite dihedral group
8. is a torsion free arithmetic groupy
9. is a Bieri-Eckmann duality group
10. = Zd * Zm
An application to group cohomology. The existence of the norm
map enables one to define a version of Farrell-Tate cohomology for an
arbitrary group and an arbitrary G-spectrum:
Definition 1.3. Let EtG be the homotopy cofiber of the norm map
DG ^hG E ! EhG :
6 JOHN R. KLEIN
Define the generalized Farrell-Tate cohomology of G with coefficients in
E to be the homotopy groups of EtG:
bE*(G) := ss- *(EtG) :
Remark 1.4. If G is discrete and and M is a G-module then we re-
cover the case of Tate cohomology by taking E = HM (the Eilenberg-
Mac Lane spectrum of M). More generally, Farrell-Tate cohomology
theory is defined when G is discrete and has finite virtual cohomological
dimension (cf. [Br4 , Ch. X]).
In another paper [Kl1 ] we will show that the homotopy groups of
the cofiber of the norm map coincide with the Farrell-Tate groups in
two instances: (a) When G is finite (i.e., the Tate case), or (b) when
M admits a finite type projective resolution over Z[G].
By the Theorem D, Eb*(G) = 0 if BG is finitely dominated. What
happens if G has a subgroup H of finite index such that BH is finitely
dominated? The following result shows that in high degrees one re-
covers the group cohomology of G with coefficients in the spectrum E.
Define E*(G) to be ss- *(EhG ). Then
Theorem E. Assume that E is connective (i.e., (- 1)-connected). Sup-
pose that G admits a subgroup H of finite index such that
o BH is finitely dominated.
o H*(BH; M) = 0 for * > n and any local coefficient bundle M on
BH.
Then there is an isomorphism
bE*(G) ~=E*(G) if* > n :
If G is discrete, then in the language of group cohomology, the hy-
potheses of the theorem amount to saying that G is VFP and has virtual
cohomological dimension n. In the classical situation when E = HZ
and G is discrete, the theorem specializes to one of the well-known
properties of Farrell-Tate cohomology.
Applications to Poincare duality spaces. Suppose that F ! E !
B is a fibration of connected, finitely dominated spaces. Choose a
basepoint for F . Applying a suitable group model for the loop space,
we obtain an extension of topological groups
1 ! F ! E ! B ! 1 :
(Details: let .E and .B denote the Kan loop groups of the total
singular complex of E and B. Define .F to be the kernel of the onto
homomorphism .E ! .B. Then apply realization.)
THE DUALIZING SPECTRUM 7
Since the smash product of two spectra is weak equivalent to the
sphere spectrum if and only if each constituent is,1 it follows that DE
is a sphere if and only if DF and DB are spheres. Applying Theorem
B, we have
Corollary F. With respect to the above assumptions, E is a Poincare
space if and only if F and B are.
The corollary has a history. C.T.C. Wall first posed the statement as
a question, and a solution was announced by Quinn (unpublished, but
see [Qu2 ]). A proof involving manifold techniques was first published
by Gottlieb [Go ]. The present proof is homotopy theoretic.
Suppose that X 2 Rn is the compact regular neighborhood of a con-
nected finite polyhedron. Assume that the spine of X has codimension
3 (this can be arranged, if necessary, by embedding X in a higher
dimensional euclidean space). Let F denote the homotopy fiber of the
inclusion of the boundary @X ! X. Since @X is a closed manifold, we
have
Corollary G. The space X satisfies Poincare duality if and only if F
is homotopy finite.
The `if' part follows directly from Corollary F, whereas the `only if'
part is well-known. In fact, F has the homotopy type of a sphere in this
instance. Furthermore, this is the procedure that is usually employed
to construct the Spivak fibration (see e.g., [Br1 , I.4.1] which relies on
[Br1 , 1.4.3]; for another kind of proof of the latter, see [Kl2 ]).
Another application of the dualizing spectrum is a new proof of an
historically important theorem of W. Browder [Br2 ] concerning finite
H-spaces (where `H-space' now means `Hopf space' = space with mul-
tiplication up to homotopy):
Theorem H. A connected finitely dominated H-space satisfies Poincare
duality.
Note: Browder asserted this only for finite H-spaces and for Poincare
duality with Z-coefficients, so we are actually asserting more. The idea
of the new proof runs as follows: if X is a finitely dominated H-space,
____________
1Proof: Suppose that X ^Y ' S0, and that X and Y are CW -spectra. By the
K"unneth formula, it is sufficient to show that X and Y are homotopy finite spe*
*ctra.
Since we have a weak equivalence of hom-spaces hom (X; -) ' hom(S0; Y ^ -), we
may infer that hom (X; -) commutes with colimits. If we write X as a colimit of
finite CW spectra, it follows that the identity map of X factors up to homotopy
through some finite spectrum. We infer that X is homotopy finite. Similarly, so*
* is
Y . I wish to thank T. Goodwillie and S. Schwede for showing me this argument.
8 JOHN R. KLEIN
then we shall prove that the dualizing spectrum of (a topological group
model for) its loop space X is unequivariantly homotopy finite. Then
the claim that X is a Poincare duality space follows from Theorem A.
Observe the similarity of Addendum C with what happens in the
smooth case: if p :E ! B is a smooth submersion of compact mani-
folds, then we have the splitting oE = oEfib p*oB where oEfibis the tan-
gent bundle along the fibers. Equivalently, in terms of stable normal
bundles E = fibE p*B . This last fact is the analogue of our adden-
dum if we regard DG together with its G-action as an object akin to
the stable normal bundle. This connection will be made precise in 5.1
and in x10. In the case of Poincare spaces, we will prove:
Theorem I. Let F ! E ! B be a fibration of connected finitely
dominated spaces. If E is a Poincare space then its Spivak fibration E
has the stable fiber homotopy type of a fiberwise join
fibF*B p*B ;
where fibFis a certain prolongation of the Spivak fibration of F to a
spherical fibration over E and p*B denotes the pullback of the Spivak
fibration of B to E.
Yet another consequence of our machinery is a result which says fi-
brations of connected finitely dominated spaces admit fiberwise Poincare
space thickenings:
Theorem J (Fiber Poincare Thickening). Let F ! E ! B be a fibra-
tion of connected finitely dominated spaces. Then there is a fibration
pair
(F 0; @F 0) ! (E0; E00) ! (B; B)
such that F 0! E0 ! B is fiber homotopy equivalent to F ! E ! B
and (F 0; @F 0) is a Poincare space.
A note on methods. This paper relies heavily on the paper [Kl5 ].
The proofs of the results listed above are homotopy theoretic. There is
only one place in the paper where a manifold argument appears: in the
appendix, in order to identify the dualizing spectrum of a compact Lie
group, we use the exponential map (cf. 10.1). This result is then used
to show that the norm map is a weak equivalence for induced spectra
(cf. 10.2).
Outline. x2 is primarily language and basic homotopy invariant con-
structions which can be applied to equivariant spectra. x3 is about
the proof of Theorem D. In x4, we prove Theorem B and Addendum
THE DUALIZING SPECTRUM 9
C. x5 contains the proof of Theorem A. In x6 we prove Theorem H.
The proof of Theorem I is in x7. In x8 we prove Theorem J, and in
x9 we prove Theorem E. x10 is the appendix, in which we identify the
dualizing spectrum of different kinds of groups, and end the discussion
with a problem, a question and a conjecture.
Acknowledgements. I would like to thank Tom Goodwillie, Randy Mc-
Carthy and Andrew Ranicki for suggestions that lead to improvements
in the exposition. Thanks also to Ross Geoghegan, who told me where
to look in the literature to find answers to some of my questions.
I originally thought of the dualizing spectrum as a gadget assigned
to a topological space (i.e., X 7! DX ). I am grateful to Greg Arone,
who suggested that my norm map should be related to the classical
norm map in the compact Lie case. Although his suggestion is not
verified in this paper, it did lead to thinking of the dualizing spectrum
as a gadget assigned to a topological group. The latter point of view
turned out to be ultimately the more fruitful one.
2. Preliminaries
Spaces. All spaces below will be compactly generated, and Top will
denote the category of compactly generated spaces. In particular, we
make the convention that products are to be retopologized with respect
to the compactly generated topology. Let Top* denote the category
of compactly generated based spaces. A weak equivalence of spaces
is shorthand for (a chain of) weak homotopy equivalence(s). A weak
equivalence is denoted by !~, whereas, we often write chains of weak
equivalences using ' (the same notation will be used when discussing
weak equivalences of spectra). A space is homotopy finite it is weak
equivalent to a finite CW complex. It is finitely dominated if it is a
retract up to homotopy of a finite CW complex.
Homotopy colimits of diagrams of spaces are formed by applying
the total singularization functor, taking the homotopy colimit of the
resulting diagram of simplicial sets (as in [B-K ]) and thereafter applying
the realization functor.
If X is a connected based space, we associate a topological group
object G of Top as follows: let S.X denote the simplicial total singular
complex of X, and let G. denote its Kan loop group. Define G to be
the geometric realization of the underlying simplicial set of G.. The
assignment X 7! G is a functor. Moreover, there is a functorial chain
of weak homotopy equivalences connecting BG to X.
Remark 2.1. In this paper a "topological group" always means either:
(i) the realization of a simplicial group, or (ii) a Lie group.
10 JOHN R. KLEIN
Poincare spaces. A space X is a Poincare duality space of (formal)
dimension n if there exists a bundle of coefficients L which is locally
isomorphic to Z, and a fundamental class [X] 2 Hn(X; L) such that
the associated cap product homomorphism
\[X] :H*(X; M) ! Hn- *(X; L M)
is an isomorphism in all degrees. Here, M denotes any bundle of coef-
ficients.
Usually, Poincare spaces are implicitly understood to have some sort
of finiteness condition imposed upon them. For the most part, we shall
assume that X is a finitely dominated CW complex.
More generally, a CW pair (X; @X) is a Poincare pair of dimension n
if there exists a bundle of coefficients L which is locally isomorphic to Z,
and a fundamental class [X] 2 Hn(X; @X; L) such that the associated
cap product homomorphism
\[X] :H*(X; M) ! Hn- *(X; @X; L M)
is an isomorphism in all degrees for all local coefficient bundles M, and
furthermore, @*[X] 2 Hn- 1(@X; L|@X) equips @X with the structure of
a Poincare duality space. See [Wa2 ] or [Wa3 ] for more details.
Spectra. A spectrum will be taken to mean a collection of based spaces
{Xi}i2N together with based maps Xi ! Xi+ 1where Xi denotes the
reduced suspension of Xi. A map of spectra X ! Y consists of maps
Xi ! Yi which are compatible with the structure maps.
Let G be a topological group. A (naive) G-spectrum consists of
a spectrum X such that each Xi is a based (left) G-space and each
structure map Xi ! Xi+ 1is equivariant, where the action of G on
Xi is defined so as to act trivially on the suspension coordinate.
Homotopy groups are defined in the usual way. Maps of G-spectra
are maps of spectra that are compatible with the G-action. Let SpG
denote the category of these. One way to obtain a G-spectrum is to
take a based G-space X and form its suspension spectrum 1 X; the
j-th space of the latter is Q(Sj ^ X), where Q = 1 1 is the stable
homotopy functor. In particular, S0[G] is the suspension spectrum of
G+ .
A weak equivalence of G-spectra is a morphism inducing an isomor-
~G
phism on homotopy groups. Weak equivalences are indicated by ! ,
and we say that two G-spectra X and Y are weak equivalent, written
X 'G Y , if there is a finite chain of weak equivalences, starting at X
and terminating at Y .
A map of spectra is r-connected if it induces a surjection on homo-
topy up through degree r and an isomorphism in degrees less than r. A
THE DUALIZING SPECTRUM 11
spectrum is r-connected if the map to the trivial spectrum (consisting
of the one point space in each degree) is (r+ 1)-connected. A spectrum
is bounded below if it is r-connected for some r.
S. Schwede has shown that the above notion of weak equivalence
arises from a Quillen model category structure on SpG (cf. [Sc ]). In
this model structure, a fibrant object is a G-spectrum X which is an
-spectrum: the adjoint Xn ! Xn+ 1to the structure maps are weak
homotopy equivalences. A cofibrant object is (the retract of) a G-
spectrum X such that Xn is built up from a point by attaching free
G-cells (i.e., Dnx G), moreover, the structure maps Xn ! Xn+ 1are
given by attaching free G-cells to Xn.
Any G-spectrum X has a (functorial) cofibrant approximation: there
exists a cofibrant G-spectrum Xc and a weak equivalence Xc ~! X (in
fact Xc can be constructed by the usual procedure of killing homo-
topy groups). Similarly, X has a (functorial) fibrant approximation:
there exists a fibrant G-spectrum Xf and a weak equivalence X !~ Xf
(this can be constructed by taking Xfnto be the homotopy colimit
hocolim jjXn+ j.)
Generally, we will assume that the collection of spaces describing
a G-spectrum are CW complexes. If the result Y of a construction
on X fails to have this property, we apply the functor Yn 7! |S.Yn|,
the realization of the singularization functor. The result gives a G-
spectrum which is degreewise a CW complex.
Smash products and functions with spaces. If U is a G-space and
X is a G-spectrum, then U ^ X will denote the G-spectrum which in
degree j is the smash product U ^Xj provided with the diagonal action.
This has the correct homotopy type if the underlying space of U is a
CW complex. (Here and elsewhere, we say that a construction gives
the "correct homotopy type" if it respects weak equivalences. Thus,
the functor U 7! U ^ X respects weak equivalences whose domain and
codomain are CW complexes.)
Give U ^ X the diagonal G-action. Then we can from the orbit
spectrum
U ^G X
given by taking G-orbits degreewise. In general, the latter has the
correct homotopy type if U is a based G-CW complex which is free
away from the basepoint.
Similarly, we can form the function spectrum F (U; X) which in de-
gree j is given by F (U; Xj) = the function space of unequivariant based
maps from U to Xj. An action of G on F (U; X) provided by conju-
gation (i.e., (g * f)(u) = gf(g-1 u) for g 2 G and f 2 F (U; Xj)). In
12 JOHN R. KLEIN
general, for F (U; X) to have the correct homotopy type, it is necessary
to assume that X is fibrant and that the underlying space of U is a
CW complex.
Let
F (U; X)G
denote the fixed point spectrum of G acting on F (U; X) , i.e., the spec-
trum whose j-th space consists of the equivariant functions from U to
Xj. The fixed point spectrum has the correct homotopy type if X is
fibrant and U is a based G-CW complex which is free away from the
base point.
In what follows below we sometimes abuse notation: if X fails to be
fibrant (but U is a based G-CW complex which is free away from the
basepoint), we take F (U; X)G to mean F (U; Xf)G .
Smash products of equivariant spectra. We will not require inter-
nal smash products of spectra which are strictly associative, commuta-
tive and unital. However, we will require that these have been defined
so as to be homotopy associative, commutative and unital.
In particular, a naive type construction will suffice for our purposes:
if X is a G-spectrum and if Y is an H-spectrum then X ^ Y is the
(Gx H)-spectrum whose (2n)-th space is Xn ^ Yn and whose (2n+ 1)-st
space is Xn+ 1^Yn. If H = G, then G acts diagonally on X ^Y . We can
then form the associated orbit spectrum X ^G Y . This has the correct
homotopy type provided that X or Y is cofibrant.
Suppose X; Y and Z are spectra, and that we are given maps fij:Xi^
Yj ! Zi+ jcompatible with the structure maps of X, Y and Z. Then
we obtain a map of spectra X ^ Y ! Z.
Homotopy orbits and homotopy fixed points. If X is a G-spectrum
then the homotopy orbit spectrum XhG is the (non-equivariant) spec-
trum given by
X ^G EG+ ;
where EG is the free contractible G-space (arising from the bar con-
struction), and EG+ is the result of adding a basepoint to EG.
The homotopy fixed point spectrum XhG is given by
F (EG+ ; X)G :
(recall that our conventions specify F (EG+ ; X)G to mean F (EG+ ; Xf)G
whenever X fails to be fibrant).
THE DUALIZING SPECTRUM 13
Lemma 2.2. Let X be a bounded below G-spectrum. For any Z[ss0(G)]-
module M, let HM denote the corresponding Eilenberg-Mac Lane spec-
trum, with G acting by means of the homomorphism G ! ss0(G).
Suppose that the homotopy orbit spectrum
X ^hG HM
is r-connected for every M. Then X is r-connected.
Proof. First assume that G is connected. It is shown in [Kl2 , Lemma
1.3] that if G is connected, X is bounded below and XhG is weakly
contractible, then X is also weakly contractible. In proving this we
actually showed the stronger statement that if XhG is r-connected,
then X is r-connected. The Hurewicz theorem (for bounded below
spectra) shows that XhG is r-connected if X ^ HZ is r-connected. This
gives the result when G is connected.
When G isn't connected, we can reduce to the connected situation
as follows: notice that X ^hG HM coincides up to homotopy with
XhG0 ^hss0(G)HM, where G0 is the kernel of G ! ss0(G). Take M
to be Z[ss0(G)]. It follows that (HZ ^ X)hG0 is r-connected. But G0
is connected. Therefore HZ ^ X is also r-connected by the previous
paragraph. The Hurewicz theorem now enables one to conclude that __
X is r-connected. |__|
Homotopy invariance of DG . Suppose that H.! G.is a monomor-
phism of simplicial groups. Taking realization we get a closed monomor-
phism H ! G of topological groups. Then DG is also an H-spectrum
by restriction.
Lemma 2.3. With respect to the above hypotheses, assume in addition
that H ! G induces an isomorphism on homotopy groups. Then there
is an equivariant weak equivalence
DH 'H DG :
Proof. Note that EG also serves as a model for EH. The equivariant
weak equivalence of dualizing spectra is given by the chain
~H 0 H ~H 0 H
F (EG+ ; S0[G])G ! F (EG+ ; S [G]) F (EG+ ; S [H]) :
where the first map is the inclusion of G-fixed sets into H-fixed sets_
and the second map is induced by the inclusion S0[H] ! S0[G]. |__|
Induced spectra. Let H ! G is a homomorphism, and let X be
an H-spectrum. Then one may form the induced spectrum, the G-
spectrum given by
X ^H G+ ;
14 JOHN R. KLEIN
where the action in degree j is defined by
g * (x; fl) := (x; flg-1 ) ;
with g 2 G; fl 2 G+ ; x 2 Xj.
Now assume that H G is the closed inclusion of a normal sub-
group. Let Q = G=H. If X happens to be a G-spectrum to begin with,
then the induced spectrum X ^H G+ comes equipped with (Gx Q)-
action: the action of Q is defined by
k * (x; fl) := (^kx; ^kfl)
for k 2 Q; x 2 Xj; fl 2 G+ and ^k2 G denoting any representative lift
of k.
For g 2 G, let g 2 Q denote its image. Let g 2 G act on Q+ by the
rule g * x = x(g)-1. If Z is a G-spectrum, give Z ^ Q+ the associated
diagonal action.
The following is probably well-known.
Lemma 2.4. Assume that X is a G-spectrum. Then there is a weak
equivalence
X ^H G+ 'Gx Q X ^ Q+ :
In particular, taking H = G, there is a weak equivalence
X ^G G+ 'G X :
Proof. If Y is a G-space then there is an homeomorphism of G-spaces
Y ^H G+ ~= Y ^ Q+
defined by (y; g) 7! (g-1 y; g). This map of spaces extends to the spec-_
trum level to define the equivalence. |__|
Coinduced spectra. If H G is a closed subgroup, and E is a
(fibrant) H-spectrum, then we can form the G-spectrum
F (G+ ; E)H
This is the effect of coinducing E with respect to the inclusion H ! G.
G acts on F (G+ ; E)H by (g * OE)(x) = OE(g-1 x), where g 2 G and
OE :G+ ! Ej.
If G=H is discrete, then F (G+ ; E)H can be rewritten as the cartesian
product Y
E :
G=H
Similarly, the induced spectrum E ^H G+ may be rewritten as a wedge
_
E :
G=H
THE DUALIZING SPECTRUM 15
Now if H has finite index in G, it follows that the inclusion of the
wedge into the product is a weak equivalence. Consequently, there is
an equivariant weak equivalence
E ^H G+ 'G F (G+ ; E)H
provided that H has finite index in G (compare with the `linear ana-
logue' [Br4 , Prop. 5.9]).
Lemma 2.5. Suppose that H G has finite index. Then there is an
unequivariant weak equivalence
DH ' DG :
Proof. We have
DG = F (EG+ ; S0[G])G ' F (EG+ ; S0[H] ^H G+ )G :
Replacing the induced spectrum S0[H] ^H G+ by the coinduced spec-
trum F (G+ ; S0[H])H we obtain
DG ' F (EG+ ; F (G+ ; S0[H])H )G :
Taking the adjunction, we have that
F (EG+ ; F (G+ ; S0[H])H )G = F (EG+ ^ G+ ; S0[H])Hx G :
Note that H acts only on the G+ factor of EG+ ^ G+ , whereas G acts
diagonally. The second factor projection EG+ ^ G+ ! G+ is therefore
a (Gx H)-equivariant weak equivalence. But G+ isn't (Gx H)-free; we
can make it (Gx H)-free at the expense of smashing with EH+ (with
the trivial G-action and the usual H-action). This entitles us to replace
EG+ ^ G+ with EH+ ^ G+ in the function spectrum. Consequently,
DG ' F (EH+ ^ G+ ; S0[H])Hx G = F (G+ ; F (EH+ ; S0[H])H )G = DH :
__
|__|
Extending the action. Let
1 ! H ! G ! Q ! 1
denote an extension.
Consider the restriction map
~H 0 H 0
DH := F (EH+ ; S0[H])H F (EG+ ; S [H]) := DH
which is induced by inclusion EH ! EG. Since EG+ is a G-space, we
can let G act on D0H by means of the formula
g * OE = (s 7! gOE(g-1 s)g-1 ) ;
where g 2 G; OE 2 F (EG+ ; Q(Sj ^ H+ ))H and s 2 EG+ .
16 JOHN R. KLEIN
This requires some explanation: G acts on Q(Sj ^ H+ ) by conjuga-
tion on H (this makes sense, since H is normal in G). Moreover, notice
that if g 2 H, then the H-equivariance of OE gives
g * OE = (s 7! OE(s)g-1 ) :
Consequently, the G-action we have defined on D0H actually extends
the naturally given H-action. Summarizing, we have
Lemma 2.6. The map D0H! DH of H-spectra is a weak equivalence.
Moreover, the H-action on D0H extends to a G-action in a canonical
way.
3. The norm map
In this section we prove Theorem D, except in the case when G is a
compact Lie group and E is induced. That case is handled separately
in 10.2 below.
Construction of the norm map. The task is to construct a weak
map
DG ^hG E ! EhG
which is natural in E. By applying fibrant and cofibrant replacement
to E, we can assume without loss in generality that E is fibrant and
cofibrant. Then it suffices to define a map
DG ^G E ! EhG
(where the domain now has orbits instead of homotopy orbits).
Recall once again that S0[G] has a (Gx G)-action, i.e., a pair of
commuting G-actions. In order to differentiate between them, we let
G` denote the subgroup Gx 1 and Gr the subgroup 1x G. Thus DG :=
F (EG+ ; S0[G])G` is a Gr-spectrum. Similarly, we let *r denote the Gr
action and *` the G`-action.
For integers j; k 0, define a map
Nj;k:F (EG+ ; S0[G]j)G` ^Gr Ek ! (S0[G]j ^Gr Ek)hG`
by the rule
(x; e) 7! (v 7! (x(v); e))
for x 2 F (EG+ ; S0[G]j)G` and e 2 Ek. This is well defined: if g 2
G, then we have (x; e) ~ (g *r x; ge), where (g *r x)(v) = x(v)g-1 .
But Nj;k(g *r x; ge) is the function v 7! (x(v)g-1 ; ge) ~ x(v). Hence,
Nj;k(x; e) = Nj;k(g *r x; ge). Therefore, Nj;kis invariant under the
Gr-action.
THE DUALIZING SPECTRUM 17
We still need to check that Nj;kmaps into the homotopy fixed set.
That is, we must show that function v 7! (x(v); e) is G`-equivariant.
If g 2 G`, then we calculate
Nj;k(x; e)(gv) := (x(gv); e) = (gx(v); e) =: g*`(x(v); e) = g*`(Nj;k(x; e)(v)) :
Consequently, we land in the homotopy fixed set.
The map Nj;kjust constructed is compatible with the indices as j
and k vary. Hence, we obtain a map of spectra
N :DG ^G E ! (S0[G] ^Gr E)hG :
On the other hand, there is a natural identification of G-spectra
E 'G S0[G] ^Gr E ;
so we may consider N as a weak map
DG ^hG E ! EhG :
This completes the construction of the norm map.
Remark 3.1. There is a more straightforward way to think of the con-
struction, provided one is willing to admit that homotopy category of
G-spectra has internal function objects. The norm map may then be
defined as the composition pairing
hom (S0; S0[G]) ^S0[G] hom (S0[G]; E) ! hom (S0; E) ;
where hom is taken in the homotopy category of G-spectra.
We are now ready to establish the properties of the norm map.
The case when G is arbitrary and E is G-finitely dominated.
By 2.4, the norm map is clearly a weak equivalence when E = S0[G]
(since the target in this case is precisely DG and the norm map is
identified with the identity in this case). By (de-)suspending, the norm
map is a weak equivalence for the spectrum Sk ^ G+ , where k 2 Z is
any integer.
Suppose that E = E0 [ (Dk+1 ^ G+ ) is the result of attaching a
cell to a G-spectrum E0, and suppose that the norm map is a weak
equivalence for E0. We have a homotopy cofiber sequence of G-spectra
Sk ^ G+ ! E0 ! E :
Since cofiber sequences are up to homotopy fiber sequences, it follows
that we have an associated homotopy cofiber sequence
(Sk ^ G+ )hG ! (E0)hG ! EhG
and the five lemma shows that the norm map is a weak equivalence for
E. Hence the norm map is a weak equivalence for any G-homotopy
18 JOHN R. KLEIN
finite spectrum. Naturality, and the fact that retracts preserve weak
equivalences then shows that we get a weak equivalence for any G-
finitely dominated spectrum.
The case when BG is finitely dominated and E is arbitrary.
The procedure of killing homotopy groups shows that E can be ex-
pressed up to homotopy as a filtered homotopy colimit of G-spectra
Eff, where ff is an index and Effis a G-spectrum having a finite num-
ber of (free) cells_in particular, the norm map is a weak equivalence
for Eff. Since BG is finitely dominated, it follows that EG+ is a G-
finitely dominated based G-space (in the sense that up to equivariant
homotopy, it is a retract of a based G-space built up from a point by
attaching a finite number of (free) cells). The `small object argument'
now applies, yielding a weak equivalence of spectra
hocolim F (EG+ ; Eff)G ' F (EG+ ; E)G :
ff
From this equivalence, it is straightforward to deduce that the norm
map is a weak equivalence for E.
The partial converse. Suppose that the norm map is a weak equiv-
alence for all spectra E, and that ss0(G) is finitely presented. The task
is now to show that BG is finitely dominated. The idea of the proof
is that since DG ^hG E ' EhG , it follows that homotopy fixed points
commutes with arbitrary homotopy colimits (since homotopy orbits
does).
To proceed, we substitute for E the sphere spectrum S0 with trivial
G-action, and choose a weak equivalence
~G 0
hocolim Eff ! S
ff
in which Effis a finite G-spectrum (this is accomplished by the proce-
dure of killing homotopy groups).
Because taking homotopy fixed sets commutes with filtered homo-
topy colimits, the associated map
hocolim (Eff)hG ! (S0)hG
ff
is a weak equivalence.
Consider the equivariant map c :EG+ ! S0 which is given by col-
lapsing EG to the non-basepoint of S0. Because the displayed map is a
weak equivalence, there exist an index ff and an equivariant homotopy
factorization of (the stable map associated with) c:
S0 ^ EG+ ! Eff! S0 :
THE DUALIZING SPECTRUM 19
Let G0 denote the identity component of G. Taking homotopy orbits
with respect to G0, we obtain a ss0(G)-equivariant homotopy factoriza-
tion
S0 ^ "BG+ ! EffhG0! S0 ^ "BG+ ;
where "BG = EG=G0 is the universal cover of BG (note: B"G is a model
for BG0). Take the smash product with the Eilenberg-Mac Lane spec-
trum HZ and identify the resulting Eilenberg-Mac Lane ss0(G)-spectra
with Z[ss0(G)]-chain complexes. It follows that the singular chain com-
plex of B"G+ is dominated by the homotopy finite chain complex cor-
responding to the Eilenberg-Mac Lane spectrum (with ss0(G)-action)
HZ ^ EffhG0. By a result of Wall [Wa1 ] it follows that BG is finitely_
dominated. This completes the proof of Theorem D. |__|
Remark 3.2. In proving the partial converse, note that we actually
proved more: to show that BG is finitely dominated, we only need to
assume that ss0(G) is finitely presented and that the norm map is a
weak equivalence for E = S0.
To apply Theorem D in the proof of Theorem B and Addendum C,
we will require an equivariant version of the norm map. Suppose that
1 ! H ! G ! Q ! 1
is an extension. Let E be an (Lx G)-spectrum, where L is yet another
topological group.
Consider the space F (EG+ ; Ej). This again admits a (Lx G)-action
given by
(`; g) * OE := (s 7! (`; g)OE(g-1 s)) :
for g 2 G and ` 2 L, and OE :EG+ ! Ej. Taking fixed points with
respect to H identified as the subgroup 1x H Lx G, and letting j
vary, it follows that the spectrum
F (EG+ ; E)H
comes equipped with an (Lx Q)-action.
Corollary 3.3. Assume that E is an (Lx G)-spectrum. Then there is
an (Lx Q)-equivariant (weak) map
D0H^hH E ! F (EG+ ; E)H ;
where D0H is the G-spectrum of 2.6. Furthermore, this map coincides
up to homotopy with the norm map for E considered as an H-spectrum.
Proof. We first explain how each of the spectra in the statement of
3.3 are equivariant. We have already indicated how (Lx Q) acts on
F (EG+ ; E)H . Give D0H^hH E an L-action using the given L-action
20 JOHN R. KLEIN
on E and the trivial L-action on D0H. Since D0H^ E may be given
the diagonal G-action, the homotopy orbit spectrum D0H^hH E has a
Q-action. Consequently, D0H^hH E has a (Lx Q)-action.
We next explain how the map is defined (the reader may wish at this
point to consult the construction of the norm map as given in the proof
of Theorem D). Assume without loss in generality that E is fibrant
and cofibrant.
For indices j; k 0 there is a map of spaces
N"j;k:F (EG+ ; S0[H]j)H ^H Ek ! F (EG+ ; S0[H]j ^H Ek)H
given by (x; e) 7! (v 7! (x(v); e) (this is the same formula we used to
define the norm map). By a straightforward check which we omit, N"j;k
is well-defined. We claim that N"j;kis (Lx Q)-equivariant. L equivari-
ance is clear (L behaves like a dummy variable). Let g 2 Q be any
element, and let g 2 G denote any lift of it. Equivariance with respect
to Q follows from the calculation
"Nj;k(g*(x; e))(v) = (N"j;k)((g*x; ge))(v) = (gx(g-1 v); ge) =: (g*N"j;k(x; e))*
*(v) :
Letting the indices j; k now vary, we obtain a (weak) map of (Lx Q)-
spectra
D0H^hH E ! F (EG+ ; E)H
__
Unequivariantly, this (clearly) is identified with the norm map. |__|
4. Proof of Theorem B and Addendum C
Let
1 ! H ! G ! Q ! 1
be an extension
Since DG = S0[G]hG , and Q = G=H, we have a weak equivalence of
G-spectra
(1) DG 'G (F (EG+ ; S0[G])H )hQ :
We first consider the inside term F (EG+ ; S0[G])H . Since S0[G]
has a (Gx G)-action, taking homotopy fixed points with respect to H
identified as the subgroup 1x H Gx G, and applying Corollary 3.3
together with Theorem D, we obtain a weak equivalence of (Gx Q)-
spectra
(2) F (EG+ ; S0[G])H 'Gx Q D0H^hH S0[G] 'Gx Q D0H^H G+ :
Here we are using the fact that BH is finitely dominated.
THE DUALIZING SPECTRUM 21
By 2.4 we also have a weak equivalence
(3) D0H^H G+ 'Gx Q D0H^ Q+ :
Assembling, we get a weak equivalence
(4) F (EG+ ; S0[G])H 'Gx Q D0H^ Q+ :
Take homotopy fixed points of both sides of this with respect to Q
(considered as the subgroup 1x Q Gx Q). Since we are assuming
either: (i) BQ is finitely dominated, or (ii) that BH is a finitely dom-
inated Poincare space so that D0H is a sphere,2 and therefore D0H^ Q+
is Q-finitely dominated, we are in a position to apply Theorem D and
Corollary 3.3 again to obtain weak equivalences of G-spectra
DG 'G (F (EG+ ; S0[G])H )hQ by (1)
'G (D0H^ Q+ )hQ by (4)
'G DQ ^Q (D0H^ Q+ ) by Theorem D and 3.3
'G DQ ^ D0H by 2.4.
__
This completes the proof of Theorem B and Addendum C. |__|
5. Proof of Theorem A
Assume that BG is finitely dominated.
`2 ) 3': Trivial.
`3 ) 1': This will use Theorem B and the unstable equivariant duality
theory developed in [Kl5 , x6] (see also [Kl3 ]). To explain this will
require some preparation.
Recall that if X and Y are based G-CW complexes which are free
away from the basepoint, then an equivariant duality is a map d :Sn !
X ^G Y such that the associated map of function spectra
F (X; E)G !~ F (Sn; E ^G Y ) :
is a weak equivalence for any G-spectrum E (the correspondence is
given by f 7! (f ^G idY) O d).
It is shown in [Kl5 , Th. 6.5] that for any G-homotopy finite based
free G-CW complex X, there exist an integer n 0, a G-homotopy
finite free based G-CW complex Y and an equivariant duality Sn !
X ^G Y . By a straightforward argument which we omit, if X is G-
finitely dominated, then there is a G-finitely dominated Y , an n
____________
2See `1 ) 2' in the proof of Theorem A appearing in the next section. Note t*
*hat
we aren't arguing circularly since `1 ) 2' does not use Theorem B in its proof.
22 JOHN R. KLEIN
0 and an equivariant duality Sn ! X ^G Y . By taking a suitable
suspension, we can assume that Y is simply connected.
We are now ready to proceed with the proof. Since BG is finitely
dominated (as unbased space) EG+ is G-finitely dominated. Conse-
quently, there exist an integer n 0, a G-finitely dominated Y and an
equivariant duality map
d :Sn ! EG+ ^G Y :
Passing to the stable category and n-fold desuspending, we obtain a
map of spectra
S0 ! EG+ ^G - nY
inducing a weak equivalence
F (EG+ ; E)G !~ F (S0; E ^G - nY )
for any G-spectrum E. Remember that S0[G] has a (Gx G)-action.
We may take E to be S0[G] with its (Gx 1)-action. Therefore we get a
weak equivalence
DG := F (EG+ ; S0[G])G !~ F (S0; S0[G] ^G - nY ) = - nY :
By naturality, this weak equivalence is G-equivariant. Thus we con-
clude that there is an equivariant weak equivalence
DG 'G - nY :
By assumption, DG is unequivariantly homotopy finite. From this we
infer that Y is an (unequivariant) homotopy finite space.
The pair of Borel constructions
(EGx GCY; EGx GY )
(where CY denotes the cone on Y with G acting trivially on the cone
coordinate) is a (finitely dominated) Poincare pair. Poincare duality is
a consequence of two facts: firstly, the quotient associated to the pair
is EG+ ^G Y , and the statement of equivariant duality for EG+ with
respect to the G-spectrum E = the Eilenberg-Mac Lane spectrum HM
on a Z[ss0(G)]-module M gives an isomorphism
H*(EGx GCY ; M) ~= Hn+ 1- *(EGx GCY; EGx GY ; M) :
Secondly, as the inclusion EGx GY ! EGx GCY is 2-connected, [Kl4 ,
Lemma 2.1] enables one to conclude that the pair in question is a
Poincare pair.
In particular, the boundary EGx GY is a finitely dominated Poincare
duality space, and the fibration
Y ! EGx GY ! BG
THE DUALIZING SPECTRUM 23
is a fibration of connected finitely dominated spaces. Consequently,
Corollary F shows that BG is a Poincare duality space.
`1 ) 2': If BG is a Poincare duality space of dimension n say, then it
has a Spivak fibration. If we use the method of [Kl5 ], then the Spivak
fibration is given as follows: let
Sj ! EG+ ^G Y
be an equivariant duality, where Y is G-finitely dominated and 1-
connected. It is shown in the proof of [Kl5 , Cor. C] that Y is un-
equivariantly homotopy equivalent to Sj- n, and the Spivak fibration of
BG is given by the Borel construction
Y ! EGx GY ! BG :
But we know from arguments above that there is an equivariant weak
equivalence DG 'G - jY . Consequently, DG is unequivariantly weak __
equivalent to S- n. This finishes the proof of Theorem A. |__|
In the process of proving Theorem A, observe that we actually es-
tablished more:
Corollary 5.1. Suppose that BG is a finitely dominated. Then
o DG is a suspension spectrum, i.e., there is an integer j 0 and
an equivariant weak equivalence
jDG 'G 1 Y
for some G-finitely dominated 1-connected based G-space Y .
o If furthermore BG is a Poincare space, then Y is unequivariantly
weak equivalent to a sphere and the Spivak fibration of BG is given
by the Borel construction
Y ! EGx GY ! BG :
We end this section with a corollary which shows that the property
of being a Poincare duality space is preserved with respect to taking
finite coverings.
Corollary 5.2. Suppose Xe ! X is a finite covering projection, where
eXand X are connected finitely dominated spaces. Then eXis a Poincare
duality space of dimension n if and only if X is.
Proof. We may assume without loss in generality that X = BG. Then
eX ' BH where H G has finite index and the covering map is given
by BH ! BG. By 2.5 we have DH ' DG , so DH is a sphere of __
dimension - n if and only if DG is. Now apply Theorem A. |__|
24 JOHN R. KLEIN
6. The proof of Theorem H
We first give the proof while ignoring technicalities, and thereafter
fill in the details.
Let G(X) be the topological monoid of self homotopy equivalences
of X, and let G(X; *) denote the the topological monoid of based equiv-
alences. Then there is a fibration
G(X; *) ! G(X) ! X
in which the projection from total space to base is given by the evalu-
ation map at the basepoint. Using a suitable group model for the loop
space X and G(X; *), the connecting map
X ! G(X; *)
is then a homomorphism. It can also be arranged that this map is the
inclusion of a normal subgroup (see below). It follows that there is an
equivariant weak equivalence
DX 'X D0X
in which the right side has the structure of a G(X; *)-spectrum (cf.
2.6). To avoid notational clutter we assume without loss in generality
that DX comes equipped with an extension of its X-action to a
G(X; *)-action.
Now, the fibration is classified by a map u :X ! BG(X; *) which is
null homotopic: use the H-space structure on X to get a section up to
homotopy X ! G(X) of the evaluation map. Therefore, u factorizes
as X ! CX ! BG(X; *). If we loop this factorization back we obtain
a factorization of groups
X ! CX ! G(X; *) :
where the composite coincides with the connecting map. It follows that
X-action on DX admits an extension to an action of a contractible
group.
But this implies that DX is isomorphic to a spectrum with trivial
action in the homotopy category of X-spectra: the isomorphism is
defined by the chain of weak equivalences
~X ~X triv
DX DX ^ (CX)+ ! DX ;
where DtrivXmeans DX equipped with trivial action and
o the middle term DX ^ (CX)+ is given the diagonal X-action
(X acts on (CX)+ by left translation).
o The left map is defined by projection onto the first factor of the
smash product.
THE DUALIZING SPECTRUM 25
o The right map is defined by the formula (x; t) 7! t-1x, where
x 2 (DX )j and t 2 (CX)+ .
Step 2. Since DX is X-finitely dominated, the homotopy orbit
spectrum
(DX )hX
is homotopy finite.
Since the action of X on DX is homotopically trivial, it follows
that the evident map
DX ! (DX )hX
is a coretraction: homotopical triviality of the action shows that (DX )hX
is identified up to homotopy with
DX ^ (BX)+ ' DX ^ X+ ;
and a retraction is defined by the map DX ^ X+ ! DX ^ S0 that is
given by smashing the identity of DX with the based map X+ ! S0
given by collapsing X to the non-basepoint of S0.
Since DX is a retract of its homotopy orbits, we infer that DX
is homotopy finite when considered as an unequivariant spectrum. By
Theorem A, we infer that X is a Poincare duality space. This completes
the outline of the proof.
We now proceed to fill in the details. Instead of looping the classify-
ing map u :X ! BG(X; *), we consider instead the map BG(X; *) !
BG(X). Convert this map into a Serre fibration, and call the result
BG(X; *)f! BG(X). Let us think of u now as a map X ! BG(X; *)f,
and choose a null-homotopy CX ! BG(X; *)f.
Let !. denote the functor from based spaces to simplicial groups
which assigns to a based space its total singular complex followed by
its Kan loop group. Let L.X denote the kernel of the homomorphism
!.BG(X; *)f! !.BG(X). Then we have a commutative square
!.X --- ! L.X
? ?
? ?
y y
!.CX --- ! !.BG(X; *)f:
Let C0.X denote the homotopy pushout in the model category of sim-
plicial groups of the diagram
!.CX !.X ! L.X
(see [Qu1 ]). Then C0.X is a contractible and the homomorphism L.X !
!.BG(X; *)f factors through C0.X. The realization LX := |L.X| is yet
26 JOHN R. KLEIN
another topological group model for the loop space of X, and the homo-
morphism LX ! |!.BG(X; *)f| is the inclusion of a normal subgroup.
Consequently, the dualizing spectrum DLX can we modified in its
equivariant weak homotopy type to a spectrum D0LX having an action
of |!.BG(X; *)f|. By commutativity of the above square, the action
of LX on D0LX restricts to an action of !X := |!.X|, and the action
of the latter extends to the contractible group !C0X := |!.C0.X|. We
conclude from this that the action of !X on D0LX is homotopically
trivial.
Finally, observe that !X is yet another model for the loop space of
X (in particular, B!X is homotopy equivalent to X), and that D0LX
is !X-equivariantly weak equivalent to D!X . So the action of !X
on D!X is homotopically trivial. The rest of the proof follows Step
(2) above. This completes the discussion of details and the proof of_
Theorem H. |__|
7. The proof of Theorem I
The following result will be required for the proof:
Proposition 7.1. Assume that BG is finite dimensional up to homo-
topy. Assume that W is a G-spectrum. Suppose that there exists an
unequivariant weak equivalence
W ' 1 X
where X is a finite complex. Then there exist an integer j 0, a
G-space Z and an equivariant weak equivalence
jW 'G 1 Z :
Proof. By applying fibrant and cofibrant replacement, we can assume
without loss in generality that W is fibrant and cofibrant, and that X
is a CW complex. Let Aut (W ) denote the topological monoid whose
points are (unequivariant) self-maps W ! W which are weak equiva-
lences. The action of G on W specifies a homomorphism of topological
monoids G ! Aut (W ), which upon applying classifying spaces, gives
a map
BG ! BAut (W ) :
Let 1cX be the spectrum whose j-th space is Sj^X. Then 1cX is
a cofibrant version of the suspension spectrum of X By hypothesis, we
may choose an unequivariant weak equivalence 1cX !~ W , where X
is a finite dimensional complex. Let 1c,fX be the effect of (functorially)
converting 1cX into a fibrant and cofibrant (unequivariant) spectrum.
THE DUALIZING SPECTRUM 27
We assert that the homomorphism of topological monoids
:Aut (1cX) ! Aut (1c,fX)
(given by the functor which maps a function to the map induced on
fibrant approximations) is a weak equivalence of underlying spaces. To
see this, let E(1cX; 1c,fX) be the space of weak equivalences from
1cX to 1c,fX. Then the map
:Aut (1c,fX) ! E(1cX; 1c,fX)
which is given by restricting the source is a weak equivalence of under-
lying spaces, since both the source and target of are function spaces
having the `correct' homotopy type (each function space consists of
mappings out of a cofibrant object into a fibrant object).
Also the composite
O :Aut (1cX) ! E(1cX; 1c,fX)
is the map given by including targets_it too is a weak equivalence of
spaces, since 1c has a right adjoint that preserves weak equivalences
between cofibrant objects. Consequently, the homomorphism is also
weak equivalence of spaces.
Applying classifying spaces, we obtain weak equivalences
BAut (1cX) ~! BAut (1c,fX) ' BAut (W )
where the second of these equivalences arises because the fibrant and
cofibrant spectra 1c,fX and W are homotopy equivalent.
On the other hand, if Aut *(jX) refers to the topological monoid
of based self weak equivalences of the space jX, the Freudenthal
suspension theorem says that the evident homomorphism
Aut *(jX) ! Aut (1cX)
has connectivity j- 3- dim X, where dim X denotes the dimension of
X as a CW complex. Assembling, we have maps
BAut *(jX) ! BAut (W )
whose connectivity tends to infinity as j does.
Since BG is homotopy finite dimensional, there exists an integer j
such that the map BG ! BAut (W ) factors up to homotopy through
a map
BG ! BAut *(jX) :
This means that we can construct a fibration over BG with fiber jX,
such that the fibration is equipped with a section. If we pull this
fibration back along EG ! BG, the resulting total space, call it Y , is
an (unbased) space with G-action equipped with an equivariant section
28 JOHN R. KLEIN
EG ! Y . Moreover, Y is has the unequivariant homotopy type of
jX. The mapping cone of this section yields a based G-space Z again
having the unequivariant homotopy type of jX. A tedious, albeit
straightforward, checking of definitions (which we omit) shows that
1 Z and jW are equivariantly weak equivalent. This completes the __
proof of 7.1. |__|
Proof of Theorem I. Suppose that
F ! E ! B
is a fibration of connected finitely dominated Poincare duality spaces.
Choose a basepoint for F (this gives basepoints for E and B). Let
G denote the realization of the Kan loop group of the total singular
complex of E and let Q be the realization of Kan loop group of the
total singular complex of B. Define H to be the kernel of the surjective
homomorphism G ! Q. Then we have an extension
1 ! H ! G ! Q ! 1 :
Applying the classifying space functor gives us a fibration BH !
BG ! BQ which is identified with the original fibration up to weak
equivalence.
By Addendum C we have an equivariant weak equivalence
DG 'G D0H^ DQ :
By Theorem A, these spectra are all spheres, and since the classifying
spaces BH; BG and BQ are finitely dominated, they are also homotopy
finite dimensional. Consequently, we may apply 7.1 to conclude that
there exist a based Q-space Y , a based G-space Z, an integer j 0
and equivariant weak equivalences
jDQ 'Q 1 Y and jD0H'G 1 Z :
By 5.1, EQx QY ! BQ represents the Spivak fibration of BQ. Since
the pullback of EGx GZ ! BG to BH is identified with EHx H Z !
BH, and the latter is the Spivak fibration of BH, the former is a
prolongation of the latter to BG. Consequently, we have a weak equiv-
alence
jDG 'G 1 Y ^ Z ;
where G acts diagonally on the right hand side.
By 5.1, the Borel construction EGx G(Y ^ Z) ! BG is the Spivak
fibration of BG. It is straightforward to check that this last fibra-
tion has the fiberwise stable type of the fiberwise join of the_fibrations
EGx GY ! BG and EGx GZ ! BG. |__|
THE DUALIZING SPECTRUM 29
8. Proof of Theorem J
Let F ! E ! B be a fibration of connected finitely dominated
spaces. As in the last section, we can assume that this is coming from
an extension of topological groups
1 ! H ! G ! Q ! 1
by applying the classifying space functor.
Let D0H be the dualizing spectrum of H modified as in x2 so that
it has an extension to a G-action. Then according to 5.1 there exist
an integer j 0, a 1-connected based G-space Z and an equivariant
weak equivalence
D0H'H 1 Z :
Then (EGx H CZ; EGx H Z) is a Poincare pair (details omitted; the
argument is essentially the same which is used in the proof of `3 ) 1'
in Theorem A), and the fibration pair
(EGx H CZ; EGx H Z) ! (EGx GCZ; EGx GZ) ! (BQ; BQ)
__
completes the proof. |__|
9. Proof of Theorem E
Let M be any Z[ss0(H)]-module. Since EH+ is finitely dominated,
we know and that DH is equivariantly dual to EH+ (cf. the proof of
3 ) 1 in Theorem A). Consequently, there is an isomorphism
ss-*(DH ^hH HM) ~=H*(BH; M) = 0 for * > n :
Now use 2.2 to conclude that DH is (-n - 1)-connected.
By 2.5, there is an unequivariant weak equivalence
DH ' DG :
It follows that DG is also (-n - 1)-connected. But then so is the
spectrum DG ^hG E since E is (-1)-connected. Using the homotopy
cofiber sequence
DG ^hG E ! EhG ! EtG
one infers that the map EhG ! EtG is (-n)-connected. One concludes
__
from this that bE*(G) and E*(G) are isomorphic in degrees * > n. |__|
30 JOHN R. KLEIN
10. Appendix: Examples
1. BG is a finitely dominated Poincare duality space. According
to 5.1, jDG 'G 1 Y where Y is unequivariantly a sphere. Moreover,
the Borel construction Y ! EGx GY ! BG gives the Spivak fibration.
Hence, if BG has dimension n, there is an equivariant weak equiv-
alence
DG 'G S[-n] ;
where S[-n] is the fiber of the Spivak fibration desuspended down to
degree - n.
2. The case of a compact Lie group.
Theorem 10.1. Assume that G is a compact Lie group. Then there
is an equivariant weak equivalence
DG 'G SAd G;
where the right side denotes the suspension spectrum of the one point
compactification of the adjoint representation of G.
Proof. (Sketch). Thinking unstably, for the moment we take SAd G to
mean the one point compactification of the Lie algebra g of G, with
the Gx G action on it in which Gx 1 acts trivially and 1x G acts via
the adjoint representation Ad G :G ! GL (g). Give G+ the action of
Gx G defined by (g; h) * x = gxh-1. Give the based function space
F (G+ ; SAd G) the action of Gx G defined by conjugation of functions:
(g; h) * OE(y) = Ad G(h)(OE(g-1 yh)).
Let
log: G ! SAd G
be defined as follows: choose ffl > 0 such that the exponential map
exp :g ! G is an embedding on D(ffl) = the disk of radius ffl. Identify
SAd G with D(ffl)=@D(ffl). Define log(x) to be z if exp(z) = x and z has
norm ffl, and 1 otherwise.
Then the map
ff :G+ ! F (G+ ; SAd G)
given by ff(x)(y) := log(x- 1y) is (Gx G)-equivariant. (This uses the
fact that g exp(x)g-1 = expAdG (g)(x)for all g 2 G; x 2 g.)
The adjunction map ^ff:G+ ^G+ ! SAd Gof ff is a Spanier-Whitehead
duality. (Reason: using the trivialization of the tangent bundle of G
given by left translation, F (G+ ; SAd G) is identified with the space of
sections of the fiberwise one point compactification of the tangent bun-
dle. With respect to this identification, the map ff gives the tangential
version of Atiyah duality [At ].)
THE DUALIZING SPECTRUM 31
Passing to the stable category, we infer that ff induces a (Gx G)-
equivariant weak equivalence of spectra
S0[G] 'Gx G F (G+ ; SAd G) :
Taking homotopy fixed sets with respect to 1x G, we obtain
DG = (S0[G])h(Gx 1)'G F (G+ ; SAd G)Gx 1= SAd G:
__
|__|
Corollary 10.2. Suppose that G is a compact Lie group and that W
is an unequivariant spectrum. Let E = W ^ G+ . Then the norm map
DG ^hG E ! EhG is a weak equivalence.
Proof. The proof of 10.1 shows that S0[G] and F (G+ ; SAd G) are (Gx G)-
equivariantly weak equivalent. Consequently, smashing with W we get
W ^ G+ 'Gx G W ^ F (G+ ; SAd G) :
Since G+ is a finite complex, the small object argument implies that
W ^ F (G+ ; SAd G) 'Gx G F (G+ ; SAd G^ W ) :
Therefore we get a weak equivalence
W ^ G+ 'Gx G F (G+ ; SAd G^ W ) :
Taking homotopy fixed sets with respect to Gx 1, we get
EhG = (W ^ G+ )hG 'G SAd G^ W 'G DG ^ W 'G DG ^hG E :
A (tedious) check which we omit shows that this identification coincides_
with the norm map up to homotopy. |__|
3. G is a finitely dominated topological group. By 2.5, we can
assume that G is connected. We have an extension
G ! P G ! G
given by the path fibration. By Theorem H, BG ' G is a Poincare
duality space, so Theorem B says that DG ^ DG ' DPG = S0. If
G has dimension n as a Poincare duality space, then DG ' S-n . It
follows that there is an unequivariant weak equivalence
DG ' Sn :
32 JOHN R. KLEIN
4. is a torsion free discrete cocompact subgroup of a con-
nected Lie group G. In this instance B is homotopy equivalent to
the compact closed manifold
\G=K
where K G is any maximal compact subgroup. So by Theorem A,
there is a weak equivalence
D ' Sk- n
where n = dim G and k = dim K.
5. is a finitely generated free group. Suppose that is a free
group on g generators. Let Hg be a handlebody of genus g embedded
in R3. Then B ' Hg, and 10.5 and 10.6 below show
D ' S- 3^ uBssg ;
where u denotes unreduced suspension and Bssg is the space with -
action defined as follows: let ssg be the kernel of the homomorphism
ss1(@Hg) ! ss1(Hg). Then ssg acts freely on the universal cover of
the surface @Hg. The universal cover is contractible, so a model for the
classifying space Bssg is given by taking the orbit space of the ssg-action.
The orbit space therefore inherits a -action.
Unequivariantly, it is elementary to check that D weak equivalent
to an infinite countable wedge of (- 1)-spheres.
Remark 10.3. Since finitely generated free groups are arithmetic, one
can alternatively identify the dualizing spectrum in this case by ap-
pealing to example 8 below.
6. = Pn is the pure braid group. Recall that Pn is defined to be
the fundamental group of the ordered configuration space of n points in
R2. The latter is an Eilenberg-Mac Lane space, so BPn is in particular
homotopy finite.
Forgetting the last point in a configuration defines an extension
1 ! Z*g ! Pn ! Pn- 1! 1 :
Iterated application of Theorem B now shows that
DPn ' DZ*g^ DZ*(g-1)^ . .^.DZ :
Each factor on the right side is weak equivalent to a countably in-
finite wedge of (-1)-spheres. Consequently, DPn is unequivariantly
weak equivalent to a countably infinite wedge of spheres of dimension
-1_
2g(g+ 1).
THE DUALIZING SPECTRUM 33
7. = Z2* Z2 is the infinite dihedral group. It is well-known that
the infinite dihedral group has an infinite cyclic (normal) subgroup of
index 2. Consequently 2.5 shows that
DZ2*Z2' DZ ' S-1
(the last equality is a consequence of Theorem A and the fact that
BZ = S1).
8. is a torsion free arithmetic group. It is known by work
of Borel and Serre [B-S ] that there is a model for B which can be
compactified to a compact manifold with corners.
Namely, the space X := G(R)=K is a model for E, where G(R)
is the group of real points of the algebraic group where lives, and
K is a choice of maximal compact subgroup. Borel and Serre define a
manifold with (free) -action X by adding corners to X in a suitable
way. The compactification of B is then
Y = X = :
The space X is gotten from X by adjoining a `partial' boundary @X
which has the -equivariant homotopy type of = the Solomon-Tits
building of the group of rational points of G (see [Br3 , Chap. 7] for
more details).
Let So denote the Thom space of the tangent bundle of X . As
X is contractible, So is a sphere having the same dimension as X .
Moreover, So comes equipped with a based -action. Then unreduced
Borel construction So ! Sox X ! Y has the fiber homotopy type of
the fiberwise one point compactification of the tangent bundle of Y .
Using 10.5 and 10.6 below, we infer
Theorem 10.4. There is an equivariant weak equivalence
D ' F (So; 1 u) ;
where u is the unreduced suspension of the Solomon-Tits building
. In particular, up to an orientation character, the homology of D
coincides with the Steinberg representation.
Since is homotopy equivalent to wedge of spheres, unequivariantly,
D is a wedge of spheres.
Examples 5 and 8 made use of the following result:
Proposition 10.5. Let G be a topological group. Assume that BG
comes equipped with a weak equivalence h :Y !~ BG in which (Y; @Y )
is a finitely dominated Poincare pair. Let So denote the the fiber to
Spivak tangent fibration of Y (dimension shifted so that it has degree
34 JOHN R. KLEIN
n), considered as a G-spectrum. Then there is an equivariant weak
equivalence
DG 'G F (So; 1 eY=@Ye) ;
where (Ye; @Ye) denotes the fiber product (Y xBG EG; @Y x BGEG).
Remark 10.6. Note that eY is weakly contractible, so eY=@Ye is equiv-
ariantly weak equivalent to u@Ye, the unreduced suspension of @Ye.
Proof of 10.5.(Sketch). Let p :(E; E|@Y) ! (Y; @Y ) be the Spivak
normal fibration. By taking fiberwise join with S0 if necessary, we
can assume that p comes equipped with a section. The characterizing
property of the Spivak fibration is that comes equipped with a map
ff :Sj ! E =E|@Y
whose target is the Thom space of p, in which the cap product of
ff*([Sj]) with the Thom class of p is a fundamental class for (Y; @Y ).
Up to fiber homotopy equivalence, we can rewrite p as a Borel con-
struction
(S x G eY; S x G @Ye) ! (Y; @Y ) ;
where S represents the fiber of p together with its based G-action.
With respect to this identification, the Thom space E =E|@Y is iden-
tified with S ^hG eY=@Ye. In this representation, ff becomes a map
fi :Sj ! EG+ ^G (S ^ eY=@Ye) ;
and the relation between ff, the Thom isomorphism and Poincare du-
ality translates to the statement that fi is an equivariant duality map
(this also uses [Kl5 , Prop. 6.4]).
On the other hand, as in the proof of 3 ) 1 of Theorem A (see
x6), we know that there exist an integer k 0, an equivariant weak
equivalence
Sj ^ DG 'G 1 Z
and an equivariant duality map Sk ! EG+ ^G Z. By suspending if
necessary, we can assume that k = j.
By the uniqueness theorem for equivariant duals [Kl5 , Thm 6.5], we
may conclude that there is an equivariant weak equivalence
(S ^ eY=@Ye) 'G 1 Z :
Consequently, there is an equivariant weak equivalence
DG 'G S ^ eY=@Ye :
THE DUALIZING SPECTRUM 35
Since So ^ S 'G S0, we have an identification S 'G F (So; S0). Since
So is G-finitely dominated, we have F (So; S0) ^ E 'G F (So; E) for
any G-spectrum E. In particular,
DG 'G S ^ eY=@Ye 'G F (So; 1 eY=@Ye) :
__
|__|
9. is a Bieri-Eckmann duality group. Assume that B is finitely
dominated. Recall that is a duality group of dimension n if there ex-
ists a Z[]-module D such that in every degree there is an isomorphism
H*(; M) ~=Hn- *(; D Z M):
for any Z[]-module M (see [Br4 , Chap. 8, x10] for the basic properties
of duality groups). The module D is called the dualizing module of ,
and is isomorphic to Hn (; Z[]). It is known that D is torsion-free as
an abelian group. If D is finitely generated and has rank one, then B
is a Poincare space, and in this instance one says that is a Poincare
duality group.
The following result characterizes the dualizing spectra of duality
groups. We omit the proof, since it essentially follows along the lines
of the proof of Theorem A.
Theorem 10.7. A group is a duality group (of dimension n) if and
only if its dualizing spectrum D is unequivariantly weak equivalent to
a Moore spectrum in degree -n on a torsion free abelian group.
(Recall that a Moore spectrum in degree j on an abelian group A is
a spectrum Y whose spectrum homology ss*(Y ^ HZ) vanishes except
in dimension j, and whose homology in degree j is isomorphic to A.)
Remark 10.8. If is a duality group of dimension n, then it is not dif-
ficult to see that the spectrum homology of D in degree - n coincides
with the dualizing module of .
If D is not a (-n)-sphere, then it follows from Theorem A that
D is not a homotopy finite spectrum. We infer that D is a Moore
spectrum on an abelian group which is not finitely generated. Thus, we
recover a result of Farrell [Fa ] which says that the dualizing module of a
duality group is finitely generated if and only if the group is a Poincare
duality group (i.e., the dualizing module has rank one).
10. The case = Zd * Zm . Let us call a diagram
R --- ! Q
? ?
? ?
y y
P --- ! G
36 JOHN R. KLEIN
of (topological or discrete) groups an amalgamation diagram if it be-
comes homotopy cocartesian after applying the classifying space func-
tor. Associated to an amalgamation diagram, there is a homotopy
cartesian square of spectra
DG -- - ! (S0[G])hQ
? ?
? ?
y y
(S0[G])hP -- - ! (S0[G])hR :
We apply this in the following situation: let R be the trivial group,
P = Zd and Q = Zm with d; m > 0. Since P and Q are Poincare
duality groups, the square in this case becomes
DG -- - ! S- m ^P G+
? ?
? ?
y y
S- d^P G+ -- - ! S0[G] ;
where we have used Theorem D to rewrite the lower left and upper
right hand corner as homotopy orbits. Note that the action of P on
DP is trivial. Consequently S-d ^P G+ is an countably infinite wedge
of copies of S- d. Similarly, S- m ^P G+ is a countably infinite wedge of
copies of S-m , while S0[G] is a countably infinite wedge of copies of S0.
For dimensional reasons, the maps in the diagram are null homotopic.
Consequently,
_
DZd*Zm ' (S-1 _ S- d_ S- m)
I
where I is a countably infinite indexing set. In particular, Zd * Zm is
not a duality group unless d = m = 1.
A problem, a question and a conjecture. In all examples above,
DG turned out to be unequivariantly weak equivalent to a wedge of
spheres. It would be interesting to have other kinds of examples, espe-
cially in the case of a discrete group.
Based on the technique of "hyperbolization" [D-J ], Bestvina and
Mess [B-M ] have given examples of discrete groups such that B is
homotopy finite and H3(; Z[]) ~= Z=2. This implies that D is not
the homotopy type of a wedge of spheres.
Problem. Compute the homotopy type of D in the Bestvina-Mess
examples.
We know that Bieri-Eckmann duality groups are such that D
has the unequivariant weak homotopy type of a Moore spectrum on a
torsion free abelian group.
THE DUALIZING SPECTRUM 37
Question. In the case of a duality group , is the dualizing spectrum
always a wedge of spheres? (This is a rephrasing of the old question
which asks whether the dualizing module is free abelian.)
Finally, there is the issue of whether or not the unequivariant ho-
motopy type of the dualizing spectrum is a coarse invariant. If is
a finitely generated group, then the word metric equips with the
structure of a metric space.
Conjecture. Suppose that and 0 are quasi-isometric.3 Assume
that and 0 have homotopy finite classifying spaces. Then D and
D0 are unequivariantly weak equivalent.
There is positive evidence for this conjecture: with respect to our
assumptions, the spectrum homology of D coincides with H*f(E; Z),
the cohomology of E with finite supports. Gersten [Ge , Th. 8] has
shown that that H*f(E; Z) and H*f(E0; Z) are isomorphic. Conse-
quently, the spectrum homology of D and D0 are isomorphic.
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E-mail address: klein@math.wayne.edu