STABLY DUALIZABLE GROUPS
John Rognes
February 8th 2005
Abstract. We extend the duality theory for topological groups from the cl*
*assical
theory for compact Lie groups, via the topological study by (Dwyer and) J*
*.R. Klein
[Kl01] and the p-complete study for p-compact groups by T. Bauer [Ba04], *
*to a
general duality theory for stably dualizable groups in the E-local stable*
* homotopy
category, for any spectrum E. The principal new examples occur in the K(n*
*)-local
category, where the Eilenberg-Mac Lane spaces G = K(Z=p, q) are stably du*
*alizable
for all 0 q n. We show how to associate to each E-locally stably dua*
*lizable
group G a stably defined representation sphere SadG, called the dualizing*
* spectrum,
which is dualizable and invertible in the E-local category. Each stably *
*dualizable
group is Atiyah-Poincar'e self-dual in the E-local category, up to a shif*
*t by SadG.
There are dimension-shifting norm- and transfer maps for spectra with G-a*
*ction,
again with a shift given by SadG. The stably dualizable group G also adm*
*its a
kind of framed bordism class [G] 2 ss*(LE S), in degree dimE (G) = [SadG]*
* of the
P icE -graded homotopy groups of the E-localized sphere spectrum.
Contents
1.Introduction
1.1. The symmetry groups of stable homotopy theory
1.2. Algebraic localizations
1.3. Chromatic localizations
1.4. Applications
2.The dualizing spectrum
2.1. The E-local stable category
2.2. Dualizable spectra
2.3. Stably dualizable groups
2.4. The dualizing and inverse dualizing spectra
3.Duality theory
3.1. Poincar'e duality
3.2. Inverse Poincar'e duality
3.3. The Picard group
4.Computations
4.1. A spectral sequence for E-homology
______________
1991 Mathematics Subject Classification. 55M05, 55P35, 57T05.
Key words and phrases. K(n)-compact group, adjoint representation, Poincar'e*
* duality, norm
map, framed bordism class.
Typeset by AM S-T*
*EX
1
2 JOHN ROGNES
4.2. Morava K-theories
4.3. Eilenberg-Mac Lane spaces
5.Norm and transfer maps
5.1. Thom spectra
5.2. The norm map and Tate cohomology
5.3. The G-transfer map
5.4. E-local homotopy classes
References
1. Introduction
1.1. The symmetry groups of stable homotopy theory.
Compact Lie groups occur naturally as the symmetry groups of geometric ob-
jects, e.g. as the isometry groups of Riemannian manifolds [MS39]. Such geometr*
*ic
objects can usefully be viewed as equivariant objects, i.e., as a spaces with a*
*n action
by a Lie group. The homotopy theory of such equivariant spaces is quite well ap-
proximated by the corresponding stable equivariant homotopy theory, which in its
strong "genuine" form relies, already in its construction, on the good represen*
*tation
theory for actions by Lie groups on finite-dimensional vector spaces.
As a first example of a useful stable result, consider the Adams equivalence
Y=G ' ( -adG Y )G of [LMS86, II.7]. Here Y is any free G-spectrum, adG denotes
the adjoint representation of G on its Lie algebra and -adG Y is the stably de*
*fined
desuspension of Y with respect to this G-representation.
As a second example, Atiyah duality [At61] asserts that if M is a smooth clo*
*sed
manifold with stable normal bundle , the functional (Spanier-Whitehead) dual
DM+ = F (M+ , S) of M+ is equivalent to the Thom spectrum T h( # M). When
M = G is a compact Lie group, and thus parallelizable, we can write this as a
stable Poincar'e duality equivalence DG+ ' T h(ffl-n # G) = -n 1 (G+ ). But G
acts on itself both from the left and the right, and the bi-equivariant form of*
* this
equivalence takes the more precise form
DG+ ^ SadG ' 1 (G+ )
where G acts by conjugation from the left on the one-point compactification SadG
of the adjoint representation and trivially from the right. See Theorem 3.1.4 b*
*elow.
As a third example, the left-invariant framing of an n-dimensional compact L*
*ie
group G gives it an associated stably framed cobordism class [G] in frn~=ssn(S*
*),
the n-th stable stem. For example [S1] = j 2 ss1(S) realizes the stable class o*
*f the
Hopf fibration j :S3 ! S2. It is of interest to see which stable homotopy clas*
*ses
actually occur in this way [Os82].
The formulation of these three results may appear to require that G admits a
geometric representation theory, with tangent spaces, adjoint representations, *
*etc.,
but in fact much less is required, and that is the main thrust of the present a*
*rticle.
1.2. Algebraic localizations.
Homotopy-theoretically, the main properties of compact Lie groups are that
they are compact manifolds, hence admit the structure of a finite CW complex,
STABLY DUALIZABLE GROUPS 3
and that they are topological groups, hence are (homotopy equivalent to) loop
spaces. Browder [Br61, 7.9] showed that all finite H-spaces are Poincar'e compl*
*exes,
and recently Bauer, Kitchloo, Notbohm and Pedersen [BKNP04] showed that all
finite loop spaces are indeed manifolds (but not generally Lie groups [ABGP04]).
A standard method in homotopy theory, and a key ingredient in [BKNP04], is the
possibility to study homotopy types locally, say with respect to a Serre class,*
* an
algebraic localization in the sense of Sullivan and Bousfield-Kan, or a Bousfie*
*ld
localization with respect to a homology theory [Bo75], [Bo79].
In the p-complete category, where a map (of spaces or spectra) is considered
to be an equivalence if it induces an isomorphism on ordinary homology with Fp-
coefficients, the local incarnations of finite loop spaces are the p-compact gr*
*oups
of Dwyer and Wilkerson [DW94]. These are topological groups G ' BG with
(totally) finite mod p homology H*(G; Fp), such that the classifying space BG is
p-adically complete. We consider a compact Lie group G as a geometric, integral*
*ly
defined object, which can be analyzed one rational prime p at a time by way of *
*its
homotopy-theoretic, locally defined p-compact pieces, namely the p-compact grou*
*ps
(BG)^pobtained by p-completing the classifying space BG at p and looping. There
are also other more exotic examples of p-compact groups, which only exist local*
*ly
at one or more primes p, without the global, geometric origin of a compact Lie
group [DW93].
In his Ph.D.-thesis, T. Bauer [Ba04] showed that for each p-compact group G
one can produce a p-complete stable replacement for the adjoint representation
sphere SadG , for the purposes of p-complete stable homotopy theory. It suffice*
*s to
work G-equivariantly in the "naive" sense, where the objects are spectra equipp*
*ed
with a G-action, and the (weak) equivalences are G-equivariant maps that are
stable equivalences in the underlying non-equivariant category. Bauer showed th*
*at
for a p-compact group G, analogous results to the Adams equivalence and the
Atiyah-Poincar'e duality equivalences above hold, with SadG reinterpreted as t*
*he
dualizing spectrum ( 1 G+ )hG = F (EG+ , 1 G+ )G of W. Dwyer (unpublished)
and J.R. Klein [Kl01], but formed in the p-complete category. Bauer also showed
that a p-compact group G has the analogue of a framed bordism class [G] in ss*(*
*S^p).
For example, the Sullivan spheres (see Example 2.3.5) are examples of p-compact
groups, and represent the generators ff1 2 ss2p-3(S^p).
1.3. Chromatic localizations.
In stable homotopy theory it is well-known (following [Ra84]) that it is pos*
*sible to
localize much further than to the (algebraic) p-local or p-complete situations,*
* by way
of the chromatic Bousfield localizations with respect to the Morava- and Johnso*
*n-
Wilson spectra K(n) and E(n), for n 0. See e.g. [HS99]. We can therefore anal*
*yze
compact Lie groups and p-compact groups in even finer detail, focusing only on *
*the
p-primary vn-periodic parts of their homotopy theory, by working in the p-prima*
*ry
K(n)-local category. The topological groups G that have the finiteness property
that K(n)*(G) is finite in each degree will be called K(n)-locally stably duali*
*zable
groups, and among these we can single out the K(n)-compact groups as those whose
classifying space BG is a K(n)-local space. See Section 2.3 below. Again, there*
* are
now new, exotic, examples of K(n)-locally stably dualizable groups that only ex*
*ist
K(n)-locally for some (p, n), without even the intermediary origin of a p-compa*
*ct
4 JOHN ROGNES
group. The simplest, abelian, examples are provided by the Eilenberg-Mac Lane
spaces G = K(ss, q), e.g. for ss = Z=p, 0 q n [RW80], which are not p-compa*
*ct
for q 6= 0, and these are K(n)-compact for q 6= n.
In this paper we show that also for a K(n)-locally stably dualizable group G,
the dualizing spectrum SadG = LK(n) 1 (G+ )hG formed in the K(n)-local stable
category has the properties that make it a stable substitute for the adjoint re*
*p-
resentation sphere of a compact Lie group. The dualizing spectrum SadG is a
dualizable and invertible spectrum in the K(n)-local category, cf. Theorem 3.3.*
*4,
which means that it has an equivalence class [SadG ] 2 PicK(n) in the K(n)-loca*
*l Pi-
card group [HMS94]. In particular, suspending (smashing) by SadG is an invertib*
*le
self-equivalence of the K(n)-local category.
We show that there is a natural norm map
N :(X ^ SadG )hG ! XhG
for any spectrum X with G-action, which is a K(n)-local equivalence under sligh*
*tly
different conditions on X than those of the Adams equivalence. See Theorem 5.2.*
*4.
We also show that there is an (implicitly K(n)-local) natural Atiyah-Poincar*
*'e
duality equivalence
DG+ ^ SadG ' 1 G+ ,
which is G-equivariant from both the left and the right. See Theorem 3.1.4.
Finally, we combine the norm map N :BGadG = (SadG )hG ! ShG = D(BG+ )
for X = S with a bottom cell inclusion i: SadG ! BGadG and the projection
p: ShG ! S to obtain a natural map
pNi: SadG ! S ,
representing a homotopy class [G] 2 ss*(LK(n)S) in the PicK(n)-graded homotopy
groups of the K(n)-local sphere spectrum. See Definition 5.4.1. We informally
think of this as the K(n)-locally framed bordism class of G.
The results discussed up to now hold in a uniform manner in the E-local stab*
*le
category, for each fixed spectrum E and suitably defined E-locally stably duali*
*zable
groups. This is how the main body of the paper is written.
In Chapter 4 we develop calculational tools to study E-locally stably dualiz*
*able
groups, mostly particular to the case E = K(n). The group structure on G makes
H = K(n)*(G) a graded Frobenius algebra over R = K(n)* (Proposition 4.2.4),
for the R-dual H* = K(n)*(G) is a free graded H-module of rank 1. There is a
strongly convergent homological spectral sequence of Eilenberg-Moore type
E2s,t= TorHs,t(R, H*) =) K(n)-(s+t)(SadG )
(Proposition 4.1.1). It collapses at the E2-term to the line s = 0, and its du*
*al
identifies K(n)*(SadG ) with the H*-comodule primitives PH* (H) ~=Hom H (H*, R)
in Hom R(H*, R) ~= H = K(n)*(G) (Theorem 4.2.6). For example, when G =
K(Z=p, n) is viewed as a K(n)-locally stably dualizable group, it follows that
[G]: SadG ! S is an equivalence in the K(n)-local category (Example 5.4.6), so
the Atiyah-Poincar'e duality equivalence takes the untwisted form
F (K(Z=p, n)+ , LK(n)S) ' LK(n) 1 K(Z=p, n)+ .
STABLY DUALIZABLE GROUPS 5
1.4. Applications.
It is conceivable that more invertible spectra in the K(n)-local category ca*
*n be
constructed in the form SadG for K(n)-locally stably dualizable groups G, than
just the localized integer sphere spectra LK(n) dS for d 2 Z. There are no such
examples in the p-complete setting, but the K(n)-local Picard group is more sub*
*tle.
Likewise, it is conceivable that the associated homotopy classes [G] 2 ss*(LK(n*
*)S)
can realize more homotopy classes than those that appear from Lie groups and
p-compact groups. However, so far we have mostly studied the abelian examples of
K(n)-locally stably dualizable groups given by Eilenberg-Mac Lane spaces, where
this added potential is not realized. We think of these abelian groups as playi*
*ng the
analogous role of tori in the theory of compact Lie groups, and expect to devel*
*op a
richer supply of non-abelian examples in joint work with T. Bauer, cf. Remark 2*
*.3.7.
This work was partially motivated by the author's formulation [Ro:g] of Galo*
*is
theory of E-local commutative S-algebras. If A ! B is an E-local G-Galois exten-
sion there is a useful norm equivalence N :(B ^ SadG )hG ! BhG , with A ' BhG .
For finite groups G this follows as in [Kl01], but the natural generality for t*
*he the-
ory appears to be to allow topological Galois groups G that are E-locally stably
dualizable, as considered here. The constructions in Chapters 3 and 5 of the pr*
*esent
paper will find applications in the cited Galois theory.
Acknowledgments.
The author wishes to thank Dr. T. Bauer for discussions starting with [Ba04],
and leading to the present paper. Part of this work was done while the author w*
*as
a member of the Isaac Newton Institute for Mathematical Sciences, Cambridge, in
the fall of 2002. He wishes to thank the INI for its hospitality and support.
2. The dualizing spectrum
2.1. The E-local stable category.
As our basic model for spectra we shall take the bicomplete, bitensored clos*
*ed
symmetric monoidal category MS of S-modules from [EKMM97]. The symmetric
monoidal pairing is the smash product X ^ Y , the unit object is the sphere spe*
*c-
trum S, and the internal function object is the mapping spectrum F (X, Y ). We
write DX = F (X, S) for the functional dual. For a based topological space T we
write X ^ T = X ^ 1 T and F (T, X) = F ( 1 T, X) for the resulting bitensors.
Let E be any S-module. It induces the (generalized) homology theory E* that
takes an S-module X to the graded abelian group E*(X) = ss*(E ^ X). A map
f :X ! Y of S-modules is said to be an E-equivalence if the induced homomor-
phism f* :E*(X) ! E*(Y ) is an isomorphism, and an S-module Z is E-local if for
each E-equivalence f :X ! Y the induced homomorphism f# : [Y, Z]* ! [X, Z]*
is an isomorphism.
Let MS,E be the full subcategory of MS of E-local S-modules. There is a Bous-
field localization functor LE :MS ! MS,E [Bo79], [EKMM97, Ch. VIII] that comes
equipped with a natural E-equivalence X ! LE X for each S-module X (with LE X
E-local). Let DS = ~hMS be the homotopy category of MS , i.e., the stable cate-
gory, and let DS,E = ~hMS,E be the homotopy category of MS,E, i.e., the E-local
stable category. It is a stable homotopy category in the sense of [HPS97, 1.2.*
*2].
6 JOHN ROGNES
The induced E-localization functor LE :DS ! DS,E is left adjoint to the forgetf*
*ul
functor DS,E ! DS .
The E-local category MS,E inherits the structure of a bicomplete, bitensored
closed symmetric monoidal category from MS by applying LE to each construction
formed in MS . The symmetric monoidal pairing takes X and Y to LE (X ^ Y ),
and the unit object is the E-local sphere spectrum LE S. The internal function
object F (X, Y ) is already E-local when Y is E-local, hence does not change wh*
*en
E-localized. In a similar fashion the (limits and) colimits in MS,E are obtain*
*ed
from those formed in MS by applying the E-localization functor, and likewise for
tensors (and cotensors).
Example 2.1.1. We may take E = S, in which case every spectrum is S-local,
MS,S = MS and the S-local stable category is the whole stable category.
Example 2.1.2. For a fixed rational prime p and number 0 n < 1 we may take
E = E(n), the n-th p-primary Johnson-Wilson spectrum, with
E(n)* = Z(p)[v1, . .,.vn, v-1n] .
When n = 0, E(0) = HQ is the rational Eilenberg-Mac Lane spectrum and E-
equivalence means rational equivalence. In each case Ln = LE(n) is a smashing
localization, LnS is a commutative S-algebra and the E(n)-local category Ln =
MS,E(n), as studied in [HS99], is equivalent to the category MLnS of LnS-modul*
*es.
In this case the forgetful functor MS,E(n)! MS preserves the symmetric monoidal
pairing, but not the unit object.
Example 2.1.3. For each prime p and number 0 n 1 we may alternatively
take E = K(n), the n-th p-primary Morava K-theory spectrum. When n = 0,
K(0) = E(0) = HQ, as discussed above. When 0 < n < 1,
K(n)* = Fp[vn, v-1n]
is a graded field, and Kn = DS,K(n) is the K(n)-local stable category, again st*
*udied
in [HS99]. When n = 1, K(1) = HFp and E-equivalence means p-adic equiv-
alence, so MS,HFp is the category of p-complete S-modules. For 0 < n 1 the
forgetful functor to MS neither preserves the symmetric monoidal pairing nor the
unit object.
Convention 2.1.4. Hereafter we shall work entirely within the E-local category
MS,E. We refer to the objects of MS,E as (E-local) S-modules, or simply as spec*
*tra.
For brevity we shall write X ^ Y for the smash product, S for the sphere spectr*
*um
and F (X, Y ) for the function spectrum within this category. The same applies *
*to
the functional dual DX, limits, colimits, tensors and cotensors, all of which t*
*hen
take values in MS,E.
2.2. Dualizable spectra.
Following Dold-Puppe [DP80], Lewis-May-Steinberger [LMS86, III.1] observe
that in any closed symmetric monoidal category there are natural canonical maps
ae: X ! DDX, :F (X, Y ) ^ Z ! F (X, Y ^ Z) and ^: F (X, Y ) ^ F (Z, W ) !
STABLY DUALIZABLE GROUPS 7
F (X ^ Z, Y ^ W ). We follow Hovey-Strickland [HS99, 1.5] and say that a spectr*
*um
X is (E-locally) dualizable if the canonical map
:DX ^ X ! F (X, X)
(in the special case X = Z, Y = S) is an equivalence in MS,E. Lewis et al then
show [LMS86, III.1.2, 1.3]:
Lemma 2.2.1.
(1) The canonical map ae :X ! DDX is an equivalence if X is dualizable.
(2) The canonical map :F (X, Y ) ^ Z ! F (X, Y ^ Z) is an equivalence if X
or Z is dualizable.
(3) The smash product map ^ :F (X, Y ) ^ F (Z, W ) ! F (X ^ Z, Y ^ W ) is an
equivalence if X and Z are dualizable, or if X is dualizable and Y = S.
It follows that the function spectrum F (X, Y ) and smash product X ^ Y are
dualizable when X and Y are dualizable. In particular, DX is dualizable when X
is dualizable.
Example 2.2.2. For E = S, a spectrum X is dualizable if and only if it is stab*
*ly
equivalent to a finite CW spectrum [M96, XVI.7.4], i.e., if and only if X ' 1 *
* dK
for some finite CW complex K and integer d 2 Z.
Example 2.2.3. For E = K(n) with 0 n 1, Hovey-Strickland [HS99, 8.6]
show that a K(n)-local S-module X is dualizable if and only if K(n)*(X) is a
finitely generated K(n)*-module. Note that this includes the cases n = 0 with
K(0) = HQ and n = 1 with K(1) = HFp. In each case K(n)* is a graded field,
so K(n)*(X) will automatically be free.
Lemma 2.2.4. If a spectrum X is HFp-locally dualizable then LK(n)X is K(n)-
locally dualizable for each 0 < n < 1.
Proof. The Atiyah-Hirzebruch spectral sequence
E2s,t= Hs(X; sstK(n)) =) K(n)s+t(X)
shows that if H*(X; Fp) is a (totally) finite Fp-module, then K(n)*(X) is a fin*
*itely
generated K(n)*-module for each 0 < n < 1.
2.3. Stably dualizable groups.
Let G be a topological group. We write
S[G] = S ^ G+ = LE 1 G+
for the E-localization of the unreduced suspension spectrum on G, and DG+ =
F (S[G], S) = F (G+ , LE 1 S0) for its functional dual. We may always suppose
that G is cofibrantly based and of the homotopy type of a based CW-complex.
Definition 2.3.1. A topological group G is (E-locally) stably dualizable if S[*
*G] =
LE 1 G+ is dualizable in MS,E.
8 JOHN ROGNES
Lemma 2.3.2. The product G = G1 x G2 of two E-locally stably dualizable groups
is again E-locally stably dualizable.
Proof. If S[G1] and S[G2] are dualizable, then so is S[G] ~=S[G1] ^ S[G2].
For the following definition we shall also need to refer to Bousfield's homo*
*logical
localization for spaces [Bo75]. A map of spaces f :X ! Y is an E-equivalence *
*if
the induced homomorphism f* :E*(X) ! E*(Y ) is an isomorphism, and a space Z
is E-local if for each E-equivalence f :X ! Y the induced map of mapping spaces
f# : Map (Y, Z) ! Map (X, Z) is a weak homotopy equivalence.
Definition 2.3.3. An E-compact group is an E-locally stably dualizable group G
whose classifying space BG is an E-local space.
Example 2.3.4. If E = S, then G is a stably dualizable group if and only if G+
is stably equivalent to a finite CW complex, up to an integer suspension, cf. E*
*x-
ample 2.2.2. So each compact Lie group G is stably dualizable, since G itself t*
*hen
is a finite CW complex. If BG is nilpotent as a space then it is S-local, so in*
* this
case G is also an S-compact group.
Example 2.3.5. For E = HFp, a topological group G is stably dualizable if and
only if H*(G; Fp) is a (totally) finite Fp-module. The group G is HFp-compact
if and only if G ' BG is a p-compact group in the sense of Dwyer-Wilkerson
[DW94].
The loop space of the p-completed classifying space of a compact Lie group p*
*ro-
vides a standard example of a p-compact group, but there are also exotic exampl*
*es,
such as the p-compact Sullivan sphere (S2p-3 )^p = (B((Z=p)x n BZp)^p) for p
odd, and the 2-compact Dwyer-Wilkerson group DI(4) [DW93]. These only exist
locally, in the sense that they do not extend to integrally defined stably dual*
*izable
groups.
Example 2.3.6. For E = K(n), a topological group G is stably dualizable if and
only if K(n)*(G) is a finitely generated K(n)*-module.
By the calculations of Ravenel-Wilson [RW80, 11.1] for p odd, and [JW85, Ap-
pendix] for p = 2, each Eilenberg-Mac Lane space G = K(ss, q) = Bqss for ss a f*
*inite
abelian group is a stably dualizable group. The classifying space BG = K(ss, q *
*+ 1)
is K(n)-local if and only if ss is a (finite abelian) p-group and 0 q < n, he*
*nce in
all these cases G is K(n)-compact.
More generally, by [HRW98, 1.1] any topological group G with only finitely
many nonzero homotopy groups, each of which is a finite abelian p-group, has fi*
*nite
K(n)-homology, hence is stably dualizable.
Once again, compact Lie groups or p-compact groups provide examples of K(n)-
compact groups through K(n)-localization, but the Eilenberg-Mac Lane space ex-
amples above do not arise in this fashion. They are only defined in the chromat*
*ically
most local context, i.e., in the K(n)-local category, and do not extend to stab*
*ly
dualizable groups in the p-complete or integral category.
Remark 2.3.7. These examples are all abelian topological groups, and can be ex-
pected to play a similar role to that of tori in the theory of compact Lie grou*
*ps.
STABLY DUALIZABLE GROUPS 9
For non-abelian examples it is natural to look to finite Postnikov systems, as *
*in
[HRW98], or to looped localized Borel constructions of the form
G = LK(n)(EW xW BA)
where A is an abelian topological group, such as A = K(ss, q), the Weyl group W*
* is
a finite group acting on A, EW xW BA = B(W n A) is the classifying space of the
semi-direct product W n A and LK(n) denotes the Bousfield K(n)-localization of
spaces. To analyze the K(n)-homology of G it is necessary to study the converge*
*nce
properties of the K(n)-based Eilenberg-Moore spectral sequence in the path-loop
fibration of LK(n)B(W n A). This is joint work in progress with T. Bauer.
2.4. The dualizing and inverse dualizing spectra.
Let EG = B(*, G, G) be the usual free, contractible right G-space. Let X be
a spectrum with right G-action, and let Y be a spectrum with left G-action. We
define the G-homotopy fixed points of X to be
XhG = F (EG+ , X)G
and the G-homotopy orbits of Y to be
YhG = EG+ ^G Y .
In all cases G acts on EG from the right. These constructions only involve naive
G-equivariant spectra, or spectra with G-action, in the sense that no deloopings
with respect to non-trivial G-representations are involved. Each G-equivariant *
*map
X1 ! X2 that is an equivalence induces an equivalence XhG1 ! XhG2 of homotopy
fixed points, and similarly for homotopy orbits.
Definition 2.4.1. Let G be an E-locally stably dualizable group. The group
multiplication provides the suspension spectrum S[G] = LE 1 G+ with mutually
commuting standard left and right G-actions. We define the dualizing spectrum
SadG of G to be the G-homotopy fixed point spectrum
SadG = S[G]hG = F (EG+ , S[G])G
of S[G], formed with respect to the standard right G-action. The standard left
action on S[G] induces a left G-action on SadG .
Remark 2.4.2. A discrete group G of type F P (e.g. the classifying space BG is
finitely dominated) is called a duality group if H*(G; Z[G]) is concentrated in*
* a
single degree n and torsion free. The G-module D = Hn (G; Z[G]) is then called
the dualizing module of G, cf. [Br82, VIII.10]. The spectrum level construction
above is clearly analogous to this algebraic notion, and was previously conside*
*red
for topological groups by Dwyer and by J.R. Klein [Kl01, x1], and for p-compact
groups by T. Bauer [Ba04, 4.1]. In the latter case the finite domination hypoth*
*esis
on BG is usually unreasonable. Klein writes DG and Bauer writes SG for the
dualizing spectrum of G. We use D for the functional dual and S for the sphere
spectrum, so we prefer to write SadG instead, in view of the compact Lie group
example recalled immediately below. Our construction differs a little from that*
* of
Dwyer and Klein, due to our implicit E-localization.
10 JOHN ROGNES
Examples 2.4.3. (a) When G is a finite group, there is a canonical equivalence
S[G] = S ^ G+ ' F (G+ , S), so S[G]hG ' F (G+ , S)hG ~= F (EG+ , S) ' S is
naturally equivalent to the sphere spectrum.
(b) More generally, when G is a compact Lie group Klein [Kl01, 10.1] shows
that the dualizing spectrum SadG is equivalent as a spectrum with left G-action
to the suspension spectrum of the representation sphere associated to the adjoi*
*nt
representation adG of G, i.e., the left conjugation action of G on its tangent *
*space
TeG at the identity.
(c) In the case of a p-compact group G, Bauer [Ba04] shows that SadG ' (Sd)^p
for some integer d = dim pG called the p-dimension of G, and that SadG takes ov*
*er
the role of the representation sphere in the duality theory in that context. T*
*he
present paper extends some of Bauer's work to the E-local stable category.
Lemma 2.4.4. When G is abelian, the left G-action on SadG is homotopically
trivial, in the sense that it extends over the inclusion G EG to an action by*
* the
contractible topological group EG.
Proof. When G is abelian, the left and right G-actions on S[G] agree. In SadG =
F (EG+ , S[G])G the right action on S[G] is equal to the right action on EG+ , *
*which
in the commutative case factors as
EG+ ^ G+ EG+ ^ EG+ -! EG+ .
Remark 2.4.5. It can be inconvenient to study the E-homology of SadG directly
from its definition as a homotopy fixed point spectrum. We shall soon see that
this dualizing spectrum is the functional dual of another spectrum S-adG , which
we call the inverse dualizing spectrum, and which admits a computationally more
convenient construction as a homotopy orbit spectrum. Once we know that these
two spectra are indeed dualizable, and mutually dual, this provides a convenient
route to E-homological calculations.
Definition 2.4.6. Let G be a stably dualizable group. The left and right G-
actions on S[G] induce standard right and left G-actions on its functional dual
DG+ = F (S[G], S), respectively, by acting in the source of the mapping spectru*
*m.
We define the inverse dualizing spectrum S-adG of G to be the G-homotopy orbit
spectrum
S-adG = (DG+ )hG = EG+ ^G DG+
of DG+ , formed with respect to the standard left G-action. These left and rig*
*ht
actions commute, so the standard right action on DG+ induces a right G-action on
S-adG .
Proposition 2.4.7. There is a natural equivalence
SadG ' DS-adG
between the dualizing spectrum and the functional dual of the inverse dualizing
spectrum, as spectra with left G-action.
Proof. The canonical equivalence ae: S[G] ! DDG+ = F (DG+ , S) induces an
equivalence aehG of G-homotopy fixed points, from SadG to
F (DG+ , S)hG = F (EG+ , F (DG+ , S))G ~= F (EG+ ^G DG+ , S) = DS-adG .
STABLY DUALIZABLE GROUPS 11
3. Duality theory
3.1. Poincar'e duality.
Let G be a stably dualizable group. The topological group structure on G
makes S[G] a cocommutative Hopf S-algebra, with product OE: S[G] ^ S[G] !
S[G], unit j :S ! S[G], coproduct _ :S[G] ! S[G] ^ S[G], counit ffl: S[G] ! S
and conjugation (antipode) O: S[G] ! S[G], induced by the group multiplication
m: G x G ! G, unit inclusion {e} ! G, diagonal map : G ! G x G, collapse
map G ! {e} and group inverse i: G ! G, respectively.
The product OE and unit j makes S[G] an E-local S-algebra in MS,E, while
the coproduct, counit and conjugation need only be defined in the E-local stable
category DS,E.
The standard right G-action on DG+ makes DG+ a right S[G]-module. The
module action is given by the map
ff :DG+ ^ S[G] -! DG+
that in symbols takes , ^x to , *x: y 7! ,(xy). Inspired by [Ba04, x4.3], we co*
*nsider
the following shearing equivalence. Its definition is simpler than that conside*
*red by
Bauer, but the key idea is the same.
Definition 3.1.1. Let the shear map sh: DG+ ^ S[G] ! DG+ ^ S[G] be the
composite map
sh: DG+ ^ S[G] -1^_-!DG+ ^ S[G] ^ S[G] -ff^1-!DG+ ^ S[G] .
P P
Algebraically, sh: , ^ x 7! (, * x0) ^ x00where _(x) = x0^ x00.
The standard left and right G-actions on S[G] (and DG+ ) can be converted in*
*to
right and left G-actions on S[G] (and DG+ ), respectively, by way of the group
inverse i: G ! G. We refer to these non-standard actions as actions through
inverses. For example, the left G-action through inverses on DG+ is given by t*
*he
composite map
S[G] ^ DG+ -fl!~DG+ ^ S[G] -1^O-!DG+ ^ S[G] -ff!DG+ ,
=
where fl :X ^ Y ! Y ^ X denotes the canonical twist map. Algebraically, this
action takes (x, ,) to , * O(x): y 7! ,(O(x)y).
Lemma 3.1.2. The shear map sh is equivariant with respect to each of the follo*
*w-
ing three mutually commuting G-actions:
(1) The first, left G-action given by the action through inverses on DG+ a*
*nd
the standard action on S[G] in the source, and the standard action on S*
*[G]
in the target;
(2) The second, right G-action given by the action through inverses on DG+ *
*in
the source, and the action through inverses on DG+ in the target;
(3) The third, right G-action given by the standard action on S[G] in the s*
*ource
and by the standard actions on DG+ and S[G] in the target.
Each action is trivial on the remaining smash factors.
Proof. In each case this is clear by inspection.
12 JOHN ROGNES
Lemma 3.1.3. The shear map sh is an equivalence, with homotopy inverse given
by the composite map
DG+ ^S[G] -1^_-!DG+ ^S[G]^S[G] -1^O^1---!DG+ ^S[G]^S[G] -ff^1-!DG+ ^S[G] .
Proof. This is an easy diagram chase, using coassociativity of _, the fact that*
* ff
is a right S[G]-module action with respect to the product OE on S[G], the Hopf
conjugation identities OE(O ^ 1)_ ' jffl ' OE(1 ^ O)_, counitality for _ and un*
*itality
for ff.
Theorem 3.1.4. Let G be a stably dualizable group. There is a natural equivale*
*nce
DG+ ^ SadG -'! S[G] .
It is equivariant with respect to the first, left G-action through inverses on *
*DG+ ,
the standard left action on SadG and the standard left action on S[G]. It is *
*also
equivariant with respect to the second, right G-action through inverses on DG+ *
*, the
trivial action on SadG and the standard right action on S[G].
Proof. The shear equivalence sh: DG+ ^ S[G] ! DG+ ^ S[G] induces a natural
equivalence
(sh)hG :(DG+ ^ S[G])hG -'! (DG+ ^ S[G])hG
of G-homotopy fixed points with respect to the third, right G-action. Note that
this action is different in the source and in the target of sh.
There is a natural equivalence to the source of (sh)hG :
DG+ ^ SadG = DG+ ^ S[G]hG -'! (DG+ ^ S[G])hG .
To see that this map is an equivalence, consider the commutative square
DG+ ^ S[G]hG _____//_(DG+ ^ S[G])hG
'|| |'|
fflffl| ~= fflffl|
F (G+ , S[G]hG )_____//_F (G+ , S[G])hG .
The vertical maps are equivalences, because S[G] is dualizable and passage to h*
*o-
motopy fixed points respects equivalences. Hence the upper horizontal map is al*
*so
an equivalence.
There is also a (composite) natural equivalence from the target of (sh)hG :
(DG+ ^ S[G])hG -'! F (G+ , S[G])hG -'! S[G] .
The left hand map is an equivalence because S[G] is dualizable, by the same arg*
*u-
ment as above. The right hand map is the composite equivalence
F (G+ , S[G])hG ~= F (EG+ ^ G+ , S[G])G ~= F (EG+ , S[G]) -'!S[G] .
STABLY DUALIZABLE GROUPS 13
Here the middle isomorphism uses that G acts freely on G+ in the source.
The composite of these three natural equivalences is the desired natural equ*
*iva-
lence DG+ ^ SadG ! S[G]. The equivariance statements follow by inspection.
Remark 3.1.5. We call DG+ ^ SadG ' S[G] the Poincar'e duality equivalence. It
shows how S[G] is functionally self-dual, up to a shift by the dualizing spectr*
*um.
See also Remark 3.3.5. The equivariance statements in the theorem express the
standard left and trivial right G-actions on SadG in terms of the more familiar
G-actions on DG+ and S[G].
Lemma 3.1.6. Let G1 and G2 be stably dualizable groups. There is a natural
equivalence
SadG1 ^ SadG2 ' Sad(G1xG2)
of spectra with standard left and trivial right (G1 x G2)-actions.
Proof. The Poincar'e duality equivalences for G1, G2 and (G1 x G2) compose to an
equivalence
DG1+ ^ SadG1 ^ DG2+ ^ SadG2 ' S[G1] ^ S[G2]
' S[G1 x G2] ' D(G1 x G2)+ ^ Sad(G1xG2) .
It is equivariant with respect to the first, left (G1 x G2)-action that involve*
*s the
standard left action on SadG1 , SadG2 and Sad(G1xG2) , as well as with respect *
*to the
second, right (G1xG2)-action through inverses on DG1+ ^DG2+ and D(G1xG2)+ .
Taking homotopy fixed points with respect to the second, right action we obtain
the desired equivalence, which is equivariant with respect to the first, left a*
*ction.
Any equivalence is equivariant with respect to the trivial right action.
Remark 3.1.7. A similar relation SadG ' SadH ^ SadQ is likely to hold for an
extension 1 ! H ! G ! Q ! 1 of stably dualizable groups, cf. [Kl01, Thm. C],
but for simplicity we omit the then necessary discussion of how to promote SadH
to a spectrum with G-action, etc.
3.2. Inverse Poincar'e duality.
The aim of this section is to establish an inverse Poincar'e equivalence
S[G] ^ S-adG ' DG+ .
The initial idea is to functionally dualize the construction of the shear map i*
*n Sec-
tion 3.1, and to apply homotopy orbits in place of homotopy fixed points. Follo*
*w-
ing Milnor-Moore [MM65, x3], we identify the functional dual of a smash product
X ^ Y of dualizable spectra with the smash product DX ^ DY , in that order, via
the canonical equivalence
DX ^ DY = F (X, S) ^ F (Y, S) -^!'F (X ^ Y, S ^ S) = D(X ^ Y ) .
However, to form homotopy orbits we need genuine G-equivariant maps, and it is
generally not the case that a G-equivariant inverse can be found for the (weak)
equivalence displayed above. Thus some care will be in order.
14 JOHN ROGNES
Working for a moment in the E-local stable category DS,E = ~hMS,E, let
fi :S[G] -! S[G] ^ DG+
be dual to the module action map ff :DG+ ^ S[G] ! DG+ . It makes S[G] a right
DG+ -comodule spectrum, up to homotopy, where DG+ has the weakly defined
coproduct _0: DG+ ! DG+ ^ DG+ that is dual to OE. Furthermore, let
OE0:DG+ ^ DG+ -! DG+
be the (strictly defined) product on DG+ that is dual to _. The functional du*
*al
sh# of the shear map is then the composite
0
sh# :S[G] ^ DG+ -fi^1-!S[G] ^ DG+ ^ DG+ -1^OE--!S[G] ^ DG+ ,
which is an equivalence by Lemma 3.1.3 and duality.
Returning to the category MS,E, we shall now obtain G-equivariant representa-
tives for these maps.
Definition 3.2.1. Let "OE:S[G] ! F (S[G], S[G]) be right adjoint to the opposi*
*te
product map OEfl :S[G] ^ S[G] ! S[G]. Algebraically, O"E:x 7! (y 7! yx). Let
_# : F (S[G] ^ S[G], S[G] ^ S) ! F (S[G], S[G]) be given by precomposition by
_ :S[G] ! S[G] ^ S[G] and postcomposition with S[G] ^ S ~=S[G].
The dual shear map sh0: S[G] ^ DG+ ! F (S[G], S[G]) is defined to be the
composite map:
"OE^1
sh0:S[G] ^ DG+ - -! F (S[G], S[G]) ^ DG+
^-!F (S[G] ^ S[G], S[G] ^ S) -_#-!F (S[G], S[G])*
* .
'
It is equivariant with respect to the left G-action given by the standard left *
*actions
on S[G] and DG+ on the left hand side, and the left action through the standard
right action on the S[G] in the source of the mapping spectrum.
Theorem 3.2.2. The dual shear map sh0 is homotopic to the composite map
# fl
S[G] ^ DG+ -sh-!'S[G] ^ DG+ -'! F (S[G], S[G]) .
In particular, sh0 is an equivalence. On G-homotopy orbit spectra it induces an
equivalence
DG+ ' S[G] ^ S-adG .
Proof. The right action map ff factors up to homotopy as the composite
0^1
DG+ ^ S[G] -_--! DG+ ^ DG+ ^ S[G]
-1^fl-!DG ffl^1
+ ^ S[G] ^ DG+ - -! S ^ DG+ = DG+ .
STABLY DUALIZABLE GROUPS 15
Here ffl: DG+ ^ S[G] ! S is the pairing that evaluates a function on an element
in its source. Let j :S[G] ^ DG+ ! S be its functional dual, in the homotopy
category. Then the dual map fi factors up to homotopy as
S[G] ~=S ^ S[G] -j^1-!S[G] ^ DG+ ^ S[G]
-1^fl-!S[G] ^ S[G] ^ DG OE^1
+ - -! S[G] ^ DG+ .
A diagram chase then verifies that "OEis homotopic to the composite
S[G] -fi!S[G] ^ DG+ -fl!~DG+ ^ S[G] -! F (S[G], S[G]) .
= '
A similar chase shows that the diagram
0
S[G] ^ DG+ ^ DG+ _________________1^OE_________________//S[G] ^ DG+
' |fl^1| ' |fl|
fflffl| ^ _# fflffl|
F (S[G], S[G]) ^ DG+ __'__//_F (S[G] ^ S[G], S[G] ^ S)___//_F (S[G], S[G])
homotopy commutes.
Taken together, these diagrams show that fl Osh# ' sh0. Applying G-homotopy
orbits to the chain of equivalences
0 fl
S[G] ^ DG+ -sh-!'F (S[G], S[G]) -' S[G] ^ DG+
we obtain the desired chain of equivalences
0)hG
DG+ ' (S[G] ^ DG+ )hG -(sh----!'F (S[G], S[G])hG
-(-fl)hG--(S[G] ^ DG ) ' S[G] ^ S-adG .
' + hG
Proposition 3.2.3. Let G be a stably dualizable group. The dualizing spectrum
SadG and the inverse dualizing spectrum S-adG are both dualizable spectra. He*
*nce
S-adG ' DSadG
as spectra with right G-action. The inverse Poincar'e equivalence
S[G] ^ S-adG ' DG+
is equivariant with respect to the dual G-actions to those of Theorem 3.1.4: Th*
*e first
of these is the right G-action through inverses on S[G], the standard right act*
*ion
on S-adG and the standard right action on DG+ . The second is the left G-action
16 JOHN ROGNES
through inverses on S[G], the trivial action on S-adG and the standard left ac*
*tion
on DG+ .
Proof. It suffices to prove that S-adG is dualizable, in view of Proposition 2*
*.4.7
and Theorem 3.1.4. We must show that the canonical map
:DS-adG ^ S-adG -! F (S-adG , S-adG )
is an equivalence. We first check that smashed with the identity map of S[G] *
*is
an equivalence. This map factors as the composite
DS-adG ^ S-adG ^ S[G] ' DS-adG ^ DG+ -!' F (S-adG , DG+ )
' F (S-adG , S-adG ^ S[G]) -' F (S-adG , S-adG ) ^ S[G]*
* .
Here the first and third equivalences follow from the inverse Poincar'e equival*
*ence,
while the second and fourth equivalences are consequences of the dualizability *
*of
DG+ and S[G], respectively. Thus ^1S[G]is an equivalence. Since S is a retrac*
*t of
S[G], it follows that also itself is an equivalence. Hence S-adG is dualizab*
*le.
3.3. The Picard group.
The Picard group of the category of E-local S-modules was introduced by
M. Hopkins; see [HMS94].
Definition 3.3.1. An E-local S-module X is invertible if there exists a spectr*
*um
Y with X ^ Y ' S in MS,E. Then Y is also invertible. The smash product X ^ X0
of two invertible spectra X and X0 is again invertible.
The E-local Picard group PicE = Pic(MS,E) is the set of equivalence classes *
*of
invertible E-local S-modules. We write [X] 2 PicE for the equivalence class of
X. The abelian group structure on PicE is defined by [X] + [X0] = [X ^ X0] and
-[X] = [Y ], with X, Y and X0 as above.
Example 3.3.2. The only invertible spectra in MS are the sphere spectra Sd =
dS for integers d 2 Z, so PicS ~=Z. Similarly, in the p-complete category MS,H*
*Fp
the invertible spectra are precisely the p-completed sphere spectra (Sd)^pfor d*
* 2 Z,
so PicHFp ~= Z too.
Example 3.3.3. By Hopkins, Mahowald and Sadofsky [HMS94, 1.3], a K(n)-local
spectrum X is invertible if and only if K(n)*(X) is free of rank one over K(n)*.
These authors show [HMS94, 2.1, 2.7, 3.3] that for n = 1 and p 6= 2 there is a
non-split extension
0 ! Zxp-! Pic K(1)-! Z=2 ! 0
while for n = 1 and p = 2 there is a non-split extension
0 ! Zx2-! Pic K(1)-! Z=8 ! 0 .
Furthermore, they show [HMS94, 7.5] that when n2 2p - 2 and p > 2 there is an
injection ff :PicK(n) ! H1(Sn; ss0(En)x ), where En is the Hopkins-Miller commu-
tative S-algebra and Sn is (one of the variants of) the n-th Morava stabilizer *
*group.
This permits an algebraic identification of PicK(2) for p odd. The homomorphism
ff seems to have a non-trivial kernel for n = 2 and p = 2, cf. [HMS94, x6].
STABLY DUALIZABLE GROUPS 17
Theorem 3.3.4. Let G be a stably dualizable group. Then
SadG ^ S-adG ' S
so SadG and S-adG are mutually inverse invertible spectra in the E-local stabl*
*e cat-
egory. Hence the equivalence classes [SadG ] and [S-adG ] represent inverse ele*
*ments
in the E-local Picard group PicE .
Proof. The Poincar'e duality equivalence and the inverse Poincar'e equivalence *
*pro-
vide a chain of equivalences
S[G] ^ S-adG ^ SadG ' DG+ ^ S-adG ' S[G] ,
which is equivariant with respect to the standard left action on both copies of
S[G], the trivial action on S-adG and the standard left action on SadG . Taki*
*ng
G-homotopy orbits of both sides yields the required equivalence
S-adG ^ SadG ' S[G]hG ' S .
Remark 3.3.5. These results show that the shift given by smashing with SadG , as
in the Poincar'e duality equivalence, is really an invertible self-equivalence *
*of the
stable homotopy category of spectra with G-action, in that it can be undone by
smashing with S-adG ' DSadG .
Definition 3.3.6. Let the E-dimension of G be the equivalence class dim E(G) =
[SadG ] 2 PicE of the dualizing spectrum SadG in the E-local Picard group.
Example 3.3.7. For E = S the S-dimension of a compact Lie group G equals its
manifold dimension in PicS ~= Z. Similarly, for E = HFp the HFp-dimension of a
p-compact group G is the same as its p-dimension.
4. Computations
4.1. A spectral sequence for E-homology.
Suppose that the S-module E is an S-algebra. The standard left G-action ff0
on DG+ makes E*(DG+ ) = E-* (G) a left E*(G)-module via the composite action
map 0
E*(G) E*(DG+ ) ! E*(S[G] ^ DG+ ) -ff*!E*(DG+ ) .
Proposition 4.1.1. Let E be an S-algebra and let G be a stably dualizable grou*
*p.
There is a spectral sequence
E2s,t= TorE*(G)s,t(E*, E-* (G)) =) Es+t(S-adG )
converging strongly to E*(S-adG ) ~=E-* (SadG ).
Proof. This is the E-homology homotopy orbit spectral sequence, which is a spec*
*ial
case of the Eilenberg-Moore type spectral sequence [EKMM97, IV.6.4] for the E-
homology of
S-adG = EG+ ^G DG+ ~= S[EG] ^S[G]DG+ .
Here E*(S[EG]) ~=E*, E*(S[G]) ~=E*(G) and E*(DG+ ) ~=E-* (G). The duality
S-adG ' DSadG from Proposition 3.2.3 relates the abutment to the E-cohomology
of SadG .
18 JOHN ROGNES
4.2. Morava K-theories.
In this and the following section (4.3) we specialize to the case when E = K*
*(n),
for some fixed prime p and number 0 n 1. Hence stably dualizable means
K(n)-locally stably dualizable, etc.
Lemma 4.2.1. Let G be a stably dualizable group, so that H = K(n)*(G) is a
finitely generated (free) module over R = K(n)*. Then H is a graded cocommutati*
*ve
Hopf algebra over R, and its R-dual H* = K(n)*(G) = Hom R(H, R) is a graded
commutative Hopf algebra over R.
Proof. By [HS99, 8.6], a topological group G is stably dualizable if and only if
H = K(n)*(G) is finitely generated over R = K(n)*. The group multiplication
and diagonal map on G induce the Hopf algebra structure on H, in view of the
K"unneth isomorphism
~=
K(n)*(X) K(n)* K(n)*(Y ) -! K(n)*(X ^ Y )
in the case X = Y = S[G]. The identity K(n)*(G) ~=Hom R(H, R) is a case of the
universal coefficient theorem
~=
K(n)*(X) -! Hom K(n)*(K(n)*(X), K(n)*) .
This also leads to the Hopf algebra structure on H*.
Proposition 4.2.2. Let G be a stably dualizable group. Then K(n)*(SadG ) ~= dR
for some integer d, and K(n)*(S-adG ) ~= -d R.
Proof. By Theorem 3.3.4, SadG is an invertible K(n)-local spectrum with inverse
S-adG , so by the K"unneth theorem
K(n)*(SadG ) R K(n)*(S-adG ) ~=K(n)*(S) = R .
This implies that K(n)*(SadG ) and K(n)*(S-adG ) both have rank one over R. (Al-
ternatively, use Theorem 3.1.4 and the K"unneth theorem to obtain the isomorphi*
*sm
H* R K(n)*(SadG ) ~=H .
The total ranks of H* and H as R-modules are equal, and finite, so K(n)*(SadG )
must have rank one. In view of [HMS94, 1.3] or [HS99, 14.2], this also provides*
* an
alternative proof that SadG is invertible in the K(n)-local category.)
Definition 4.2.3. Let the integer d = degK(n)(G) such that K(n)*(SadG ) ~= dR
be the K(n)-degree of G. When 0 < n < 1 this number is only well-defined modulo
|vn| = 2(pn - 1).
Remark 4.2.4. The evident homomorphism deg: PicK(n) ! Z=|vn| takes the K(n)-
dimension of G to its K(n)-degree. By [HMS94, 1.3] or [HS99, 14.2] we also have
Eb(n)*(SadG ) ~= dEb(n)*, where bE(n) = LK(n)E(n). Similarly E*n(SadG ) ~= dE*n,
where En is the Hopkins-Miller commutative S-algebra. Taking into account the
action of the n-th Morava stabilizer group on E*n(SadG ) it is in principle pos*
*sible
STABLY DUALIZABLE GROUPS 19
to recover much more information about the K(n)-dimension of G than just the
K(n)-degree.
For any graded commutative ring R and R-algebra H, we may consider both H
and its R-dual H* = Hom R(H, R) as left H-modules in the standard way. Recall
from e.g. [Pa71, x4] that H is called a (graded) Frobenius algebra over R if
(1) H is finitely generated and projective as an R-module, and
(2) H and some suspension dH* are isomorphic as left H-modules.
It follows that H is also isomorphic to dH* as right H-modules, and conversely.
A (left or right) module M over a Frobenius algebra H is projective if and only*
* if
it is injective.
Proposition 4.2.4. Let G be a stably dualizable group. Then H = K(n)*(G) is a
Frobenius algebra over R = K(n)*. In particular, H* = K(n)*(G) is an injective
and projective (left) H-module. In fact, it is free of rank one.
Proof. Applying K(n)-homology to the equivalence of Theorem 3.1.4 gives an iso-
morphism
H* R dR = K(n)*(DG+ ) K(n)* K(n)*(SadG ) ~=K(n)*(S[G]) = H .
Here H acts from the left via the inverse of the second G-action, i.e., by the *
*standard
left action on H*, the trivial action on K(n)*(SadG ) = dR, and the left action
through inverses on H. We continue with the isomorphism
~=
O* :H = K(n)*(G) -! K(n)*(G) = H
induced by the conjugation O on S[G], which takes the left action through inver*
*ses
to the standard left action. Then the composite of these two isomorphisms exhib*
*its
H as a Frobenius algebra over R.
It is a formality that H* is injective as a left H-module, so the general th*
*eory
implies that it is also projective. But we can also see this directly in our ca*
*se, since
H* ~= -d H is an isomorphism of left H-modules, and the right hand side is free
of rank one and thus obviously projective.
Theorem 4.2.5. Let G be a K(n)-locally stably dualizable group. The spectral
sequence
E2s,t= TorHs,t(R, H*) =) K(n)s+t(S-adG )
collapses to the line s = 0 at the E2-term. The natural map i :DG+ ! S-adG
identifies
-d R = K(n)*(S-adG ) ~=R H H*
with the left H = K(n)*(G)-module indecomposables of H* = K(n)-* (G). Dually,
the natural map p :SadG ! S[G] identifies
dR = K(n)*(SadG ) ~=Hom H (H*, R)
with the left H*-comodule primitives in H.
20 JOHN ROGNES
Proof. The spectral sequence is that of Proposition 4.1.1 in the special case E*
* =
K(n). By Proposition 4.2.4, H* is a free left H-module of rank one, hence flat.*
* Thus
Tor Hs,t(R, H*) = 0 for s > 0, while for s = 0, Tor H0,*(R, H*) = R H H*. He*
*nce
the spectral sequence collapses to the line s = 0, and the edge homomorphism
corresponding to the inclusion i: DG+ ! EG+ ^G DG+ = S-adG is the surjection
H* = K(n)*(DG+ ) ! K(n)*(S-adG ) = R H H*. Thinking of H* as a left
H-module, these are the H-module coinvariants, or indecomposables, of H*.
Passing to duals, the projection p: SadG = F (EG+ , S[G]) ! S[G] is functio*
*n-
ally dual to the inclusion above, hence induces the R-dual injection Hom R(R H
H*, R) ! Hom R(H*, R) in K(n)-homology. Thus K(n)*(SadG ) is identified with
Hom R(R H H*, R) ~=Hom H (H*, R), sitting inside Hom R(H*, R) ~=H. The left
H-module structure on H* dualizes to a left H*-comodule structure on H. The
inclusion Hom H(H*, R) ! Hom R(H*, R) ~=H then identifies Hom H(H*, R) with
the H*-comodule primitives in H.
Remark 4.2.6. We sometimes write QH (H*) = R H H* for the left H-module inde-
composables of H*, and dually PH* (H) = Hom H (H*, R) for the left H*-comodule
primitives in H. Then K(n)*(S-adG ) ~=QH (H*) and K(n)*(SadG ) ~=PH* (H).
To be explicit, an element x 2 H ~= Hom R (H*, R) lies in Hom H(H*, R) if a*
*nd
only if (y * ,)(x) = ,(xy) equals ffl(y),(x) = ,(xffl(y)) for each y 2 H and , *
*2 H*.
Here ffl: H ! R is the augmentation. This condition is equivalent to asking th*
*at
xy = 0 for each y 2 ker(ffl), i.e., x 2 H multiplies to zero with each eleme*
*nt
in the augmentation ideal of H. So PH* (H) is the left annihilator ideal of the
augmentation ideal of H.
4.3. Eilenberg-Mac Lane spaces.
We can make the identifications in Theorem 4.2.5 explicit in the cases when
G = K(Z=p, q) is an Eilenberg-Mac Lane space. For p an odd prime the K(n)-
homology H = K(n)*K(Z=p, q) was computed by Ravenel-Wilson in [RW80, 9.2],
as we now recall:
Writing K(n)*K(Z, 2) ~=K(n)*{fim | m 0} with |fim | = 2m there are classes
am 2 K(n)*K(Z=p, 1) in degree |am | = 2m for 0 m < pn such that the Bockstein
map K(Z=p, 1) ! K(Z, 2) takes each am to fim . Let a(i)= api in degree |a(i)| *
*= 2pi
for 0 i < n. The q-fold cup product K(Z=p, 1) ^ . .^.K(Z=p, 1) ! K(Z=p, q)
takes a(i1) . . .a(iq)to a class aI 2 K(n)*K(Z=p, q), where I = (i1, . .,.iq) *
*and
|aI| = 2(pi1+ . .+.piq).
For q = 0, G = K(Z=p, 0) = Z=p is a finite group and not very special to the
K(n)-local situation. For each q > n, K(Z=p, q) has the K(n)-homology of a poin*
*t.
The intermediate cases 0 < q n are more interesting.
For 0 < q < n there is an algebra isomorphism
O pae(I)
K(n)*K(Z=p, q) ~= K(n)*[aI]=(aI ) ,
I
where I = (i1, . .,.iq) ranges over all integer sequences with 0 < i1 < . .<.iq*
* < n,
and ae(I) = s + 1 where s 2 {0, 1, . .,.q} is maximal such that the final s-term
subsequence has the form
(iq-s+1 , . .,.iq) = (n - s, . .,.n - 1) .
STABLY DUALIZABLE GROUPS 21
Equivalently, s is minimal such that iq-s < n - s - 1.
For q = n there is an algebra isomorphism
K(n)*K(Z=p, n) ~=K(n)*[aI]=(apI+ (-1)nvnaI) ,
where I = (0, 1, . .,.n - 1). Here |aI| = 2(1 + p + . .+.pn-1 ) = 2(pn - 1)=(p *
*- 1).
Proposition 4.3.1. For G = K(Z=p, q) with 0 < q < n, K(n)*(SadG ) is generated
Q pae(I)-1
over K(n)* by the product ss = IaI . Its K(n)-degree is 0 modulo 2(pn - 1*
*).
Proof. By Theorem 4.2.5 we identify K(n)*(SadG ) with the left H*-comodule prim-
itives in H, which consists of the elements of H that multiply to zero with eve*
*ry
element in the augmentation ideal of H. These are generated by the product ss
above. Its degree deg K(n)(G) |ss| can be computed by grouping together the
integer sequences with the same value of ae(I) = s + 1:
X
|ss|= 2(pi1+ . .p.iq)(pae(I)- 1)
I X
= 2(pi1+ . .+.piq-s+ pn-s + . .+.pn-1 )(ps+1 - 1)
1 i 0 s q
1<...