Generalized Artin and Brauer induction for compact
Lie groups
Halvard Fausk
Abstract
Let G be a compact Lie group. We present two induction theorems for certain*
* generalized
G-equivariant cohomology theories. The theory applies to G-equivariant K-th*
*eory KG,
and to the Borel cohomology associated to any complex oriented cohomology t*
*heory. The
coefficient ring of KG is the representation ring R(G) of G. When G is a fi*
*nite group
the induction theorems for KG coincide with the classical Artin and Brauer *
*induction
theorems for R(G).
1. Introduction
We present a generalization of the Artin and Brauer induction theorems for the *
*representation ring
of a finite group G. The generalization is in three directions. First, we give *
*an induction theory for
a general class of equivariant cohomology theories; the induction theorems appl*
*y to the cohomology
groups of arbitrary spectra, not just the coefficients of the cohomology theory*
*. Second, we extend
the induction theory from finite groups to compact Lie groups. Third, we alow i*
*nduction from
more general classes of subgroups than the cyclic subgroups. We use the followi*
*ng classes of abelian
subgroups of G characterized by the number of generators alowed: The class of t*
*he maximal tori
(n = 0), and for each n > 1 the class of all closed abelian subgroups A of G wi*
*th finite index in its
normalizer and with a dense subgroup generated by n or fewer elements.
In section 2 we collect some facts that we need about compact Lie groups. In*
* section 3 we
describe the induction and restriction maps in homology and in cohomology. The *
*induction theory
makes use of the Burnside ring module structure on equivariant cohomology theor*
*ies. The Burnside
ring is isomorphic to the ring of homotopy classes of stable self maps of the u*
*nit object 1GS0 in
the G-equivariant stable homotopy category. In section 4 we recall some alterna*
*tive descriptions of
the Burnside ring of a compact Lie group G and discuss some of their properties.
The following condition on a cohomology theory suffices to give the inductio*
*n theorems: We say
that a ring spectrum E has the 0-induction property if the unit map j: 1GS0 ! *
*E pre-composed
with a map f: 1GS0 ! 1GS0 is null in the stable G-equivariant homotopy catego*
*ry whenever the
underlying nonequivariant map f: 1 S0 ! 1 S0 is null. We say that a ring spec*
*trum E has the
n-induction property, for some n > 1, if the unit map j: 1GS0 ! E pre-composed*
* with a map
f: 1GS0 ! 1GS0 is null in the stable G-equivariant homotopy category whenever*
* the degree of
fA is 0 for all abelian subgroups A of G that is the closure of a subgroup gene*
*rated by n or fewer
elements. Let EG be a G-ring spectrum satisfying the n-induction property. Pick*
* one subgroup Ai
in each conjugacy class of the abelian subgroups of G with finite index in its *
*normalizer and with
a dense subgroup generated by n or fewer elements. Let |G|n be the least common*
* multiple of the
order of the Weyl groups of these abelian groups {Ai} (there are only finitely *
*many such subgroups
by Corollary 2.2). In section 5 we prove the following Artin induction theorem.
_______________________________________________________________________________*
*__
2000 Mathematics Subject Classification Primary 55P91, 19A22; Secondary 55P42
Halvard Fausk
Theorem 1.1. The integer |G|n times the unit element in E*Gis in the image of t*
*he induction map
iindGAi: iE0A! E0G.
Let MG be an EG-module spectrum, and let X be an arbitrary G-spectrum. There*
* is a restriction
map
Q Q
res: MffG(X) ! Eq[ iMresffAi(X) ' i,j,gMresffAi\gAjg-1(X)].
Here Eq denotes the equalizer and ff denotes the grading by a formal difference*
* of two finite di-
mensional real G-representations. The second product is over i, j and over g 2 *
*G. The maps in the
equalizer are the two restriction (composed with conjugation) maps. The Artin i*
*nduction theorem
implies the following Artin restriction theorem.
Theorem 1.2. There exists a map
Q Q
_: Eq[ iMresffAi(X) ' i,j,gMresffAi\gAjg-1(X)] ! MffG(X)
such that both the composites resO _ and _ O resare |G|n times the identity map.
The Brauer induction theorem is analogous to the Artin induction theorem. At*
* the expense
of using a larger class, {Hj}, of subgroups of G than those used for Artin indu*
*ction, we get that
the unit element of E*Gis in the image of the induction map from jE*Hj. As a c*
*onsequence the
corresponding restriction map, res, is an isomorphism. The exact statements are*
* given in section 6.
The G-equivariant K-theory KG(X+) of a compact G-CW-complex X is the Grothen*
*dieck con-
struction on the set of isomorphism classes of finite dimensional complex G-bun*
*dles on X. In
particular, when X is a point we get that KG(S0) is isomorphic to the complex r*
*epresentation ring
R(G). The restriction and induction maps for the equivariant cohomology theory *
*KG gives the usual
restriction and induction maps for the representation ring KG(S0) ~=R(G). The r*
*ing spectrum KG
satisfies the 1-induction property. The resulting induction theorems for R(G) a*
*re the classical Artin
and Brauer induction theorems. The details of this example are given in section*
* 9.
This work is inspired by an Artin induction theorem used by Hopkins, Kuhn, a*
*nd Ravenel
[HKR00 ]. They calculate the Borel cohomology associated to certain complex ori*
*ented cohomology
theories for finite abelian groups; then they used an Artin restriction theorem*
* to describe the Borel
cohomology, rationally, for general finite groups. We discuss induction and res*
*triction theorems for
the Borel cohomology associated to complex oriented cohomology theories when G *
*is a compact Lie
group in section 7. Singular Borel cohomology is discussed in section 8.
A Brauer induction theorem for the representation ring of a compact Lie grou*
*p was first given by
G. Segal [Seg68a]. Induction theories for G-equivariant cohomology theories, wh*
*en G is a compact
Lie group, has also been studied by G. Lewis [Lew96, sec.6]. He develops a Dres*
*s induction theory
for Mackey functors.
The idea to use the Burnside ring module structure to prove induction theore*
*ms goes back to
Conlon and Solomon [Con68, Sol67] [Ben95, chap.5].
2. Compact Lie groups
In this section we recall some facts about compact Lie groups, and provide a fe*
*w new observations.
We say that a subgroup H of G is topologically generated by n elements (or fewe*
*r) if there is a
dense subgroup of H generated by n elements. For example any torus is topologic*
*ally generated
by one element. By a subgroup of a compact Lie group G we mean a closed subgrou*
*p of G unless
otherwise stated. It is convenient to give the set of conjugacy classes of (clo*
*sed) subgroups of G a
2
Generalized Artin and Brauer induction for compact Lie groups
topology [tD79, 5.6.1]. Let d be a metric on G so that the metric topology is e*
*qual to the topology
on G. We give the space of all closed subgroups of G the Hausdorff topology fro*
*m the metric
dH (A, B) = supinfd(a, b) + supinfd(a, b).
a2Ab2B b2Ba2A
The metric topology on the space of closed subgroups is independent of the choi*
*ce of metric. Let
G denote the space of conjugacy classes of closed subgroups of G given the quo*
*tient topology from
the space of closed subgroups of G. The space G is a compact space with a metr*
*ic given by
d (A, B) = infdH (A, gBg-1).
g2G
The Weyl group WGH of a subgroup H in G is NGH=H. Let G denote the subspace of*
* G
consisting of conjugacy classes of subgroups of G with finite Weyl group. We ha*
*ve that G is a
closed subspace of G [tD79, 5.6.1].
We denote the conjugacy class of a subgroup H in G by (H), leaving G to be u*
*nderstood from
the context. Conjugacy classes of subgroups of G form a partially ordered set; *
*(K) 6 (H) means
that K is conjugate in G to a subgroup of H. Note that a subgroup of a compact *
*Lie group G can
not be conjugate to a proper subgroup of itself. (There are no properly contain*
*ed closed n-manifolds
of a closed connected n-manifold.)
A theorem of Montgomery and Zippin says that for any subgroup H of G there i*
*s an open
neighborhood U of the identity element in G such that all subgroups of HU are s*
*ubconjugate to H
[Bre72, II.5.6][MZ42 ].
Let K 6 H be subgroups of G. The normalizer NGK acts from the left on (G=H)K*
* . Montgomery
and Zippin's theorem implies that the coset (G=H)K =NGK is finite [Bre72, II.5.*
*7]. In particular,
if WGK is finite, then (G=H)K is finite. The Weyl group WGH acts freely on (G=H*
*)K from the
right by gH . nH = gnH, where gH 2 (G=H)K and nH 2 WGH. So |WGH| divides |(G=H)*
*K |.
The following consequence of Montgomery and Zippin's theorem is important for t*
*his paper. Let
GO denote the unit component of the group G.
Lemma 2.1. Let G be a compact Lie group. The conjugacy class of any abelian sub*
*group A of G
with finite Weyl group is an open point in G.
Proof.Fix a metric on G. By Montgomery and Zippin's theorem there is an ffl > 0*
* such that if K is
a subgroup of G and d ((A), (K)) < ffl, then K is conjugated in G to a subgroup*
* of A that meets
all the components of A. Let K be such a subgroup and assume in addition that i*
*t has finite Weyl
group. Then KO = AO since A < NGK and |WGK| is finite. Thus we have that (K) = *
*(A). Hence_
(A) is an open point in G. *
* |__|
Since G is compact we get.
Corollary 2.2. There are only finitely many conjugacy classes of abelian subgro*
*ups of G with
finite Weyl group.
Given a subgroup H of G we can extend H by tori until we get a subgroup K wi*
*th WGK finite.
This extension of H is unique up to conjugation. We denote the conjugacy class *
*by !(H). The
conjugacy class !(H) does only depend on the conjugacy class of H. Hence we get*
* a well defined
map !: G ! G. This map is continuous [FO05 , 1.2]. We say that !(H) is the G *
*subgroup
conjugacy class with finite Weyl group associated to H. One can also show that *
*the conjugacy class
!(H) is the conjugacy class (HT ) where T is a maximal torus in CGH [FO05 , 2.2*
*]. This result
implies the following.
Lemma 2.3. The map !: G ! G sends conjugacy classes of abelian groups to conj*
*ugacy classes
of abelian groups.
3
Halvard Fausk
We now define the classes of abelian groups used in the Artin induction theo*
*ry.
Definition 2.4. Let AG denote the set of all conjugacy classes of abelian subgr*
*oups of G with
finite Weyl group. Let AnG denote the set of conjugacy classes of abelian subgr*
*oups A of G that
are topologically generated by n or fewer elements and that have a finite Weyl *
*group. We let A0G
be the conjugacy class of the maximal torus in G.
We often suppress G from the notation of AnG and write An. We have that AnG *
*= AG for
some n by Corollary 2.2.
Example 2.5. The topologically cyclic subgroups of G are well understood. They *
*were called Cartan
subgroups and studied by G. Segal in [Seg68a]. The following is a summary of so*
*me of his results: All
elements of G are contained in a Cartan subgroup. An element g in G is called r*
*egular if the closure
of the cyclic subgroup generated by g has finite Weyl group. The regular elemen*
*ts of G are dense in
G. Two regular elements in the same component of G generate conjugate Cartan su*
*bgroups. The
map S 7! GOS=GO gives a bijection between conjugacy classes of Cartan subgroups*
* and conjugacy
classes of cyclic subgroups of the group of components G=GO. In particular, if *
*G is connected, then
the Cartan subgroups are precisely the maximal tori. The order |S=SO| is divide*
*d by |S=GO| and
divides |S=GO|2 [Seg68a, p.117]. For example the nontrivial semidirect product *
*S1oZ=2Z has Cartan
subgroups S1 and (conjugates of) 0 o Z=2Z.
Lemma 2.6. If A is a compact abelian Lie group, then it splits as
A ~=AOx ss0(A).
Proof.Since A is compact we have that ss0(A) ~= iZ=pnii. The unit component AO *
*is a torus. We
construct an explicit splitting of A ! ss0(A). Let ai2 A be an element such tha*
*t aimaps to a fixed
generator in Z=pniiand to zero in Z=pnkkfor all k 6= i. Then airaised to the pn*
*iipower maps to zero
in ss0(A), hence is in the torus AO. There is an element bi2 AO such that
ni pni
apii= bii.
Set ~ai= aib-1i, and define the splitting ss0(A) ! A by sending a generator of *
*Z=pnii2 ss0(A)_to ~ai.
Since A is commutative this gives a well defined group homomorphism. *
* |__|
The splitting in Lemma 2.6 is not natural.
Lemma 2.7. Let G be a compact Lie group, and let A 6 B be abelian subgroups of *
*G such that
!(B) is in An. Then !(A) is in An.
Proof.The minimal number of topological generators of an abelian group A is equ*
*al to the minimal
number of generators of the group of components of A by Lemma 2.6. We assume wi*
*thout loss of
generality that B has finite Weyl group. The unit component BO of B is containe*
*d in the normalizer
of A. Hence the component group of a representative for the conjugacy class !(A*
*) is isomorphic_to
a quotient of a subgroup of ss0(B). The result follows. *
* |__|
We introduce several different orders for compact Lie groups. T. tom Dieck h*
*as proved that for
any given compact Lie group G there is an integer nG so that the order of the g*
*roup of components
of the Weyl group WGH is less or equal to nG for all closed subgroups H of G [t*
*D77].
Definition 2.8. The order |G| of a compact Lie group G is the least common mult*
*iple of the orders
|WGH| for all (H) 2 G. For any nonnegative integer n let |G|n be the least com*
*mon multiple of
|WGA| for all (A) 2 AnG.
When G is a finite group all these orders coincide and are equal to the numb*
*er of elements in G.
4
Generalized Artin and Brauer induction for compact Lie groups
Remark 2.9. Let T be a maximal torus in G. Then we have that
NGT=(GO\ NGT ) ~=G=GO
since all maximal tori of G are conjugated by elements in GO. Hence the number *
*of components
|G=GO| of G divides the smallest order |G|0 = |NGT=T |. The order |G|m divides *
*|G|n for 0 6 m 6 n.
Example 2.10. For compact Lie groups the various orders might be different. An *
*example is given
by SO(3). The only conjugacy classes of abelian subgroups of SO(3) with finite *
*Weyl group are
Z=2 Z=2 and S1. The normalizers of these subgroups are (Z=2 Z=2)o 3 and S1oZ=2,*
* respectively
[tD79, 5.14]. So the Weyl groups have order 6 and 2, respectively. Hence |SO(3)*
*|n = 2 for n = 0, 1
and |SO(3)|n = 6 for n > 2. By taking cartesian products of copies of SO(3) we *
*get a connected
compact Lie group with many different orders. The order |SO(3)xN |2m is 2N 3m f*
*or m 6 N and
6N for m > N. The abelian subgroups of SO(3)xN with finite Weyl group are produ*
*ct subgroups
obtained from all conjugates of S1 and Z=2 Z=2, and furthermore all subgroups*
* of these product
subgroups so that each of the n canonical projections to SO(3) are conjugat to *
*either S1 or Z=2 Z=2
in SO(3).
3.G-equivariant cohomology theories
We work in the homotopy category of G-spectra indexed on a complete G-universe.*
* Most of the
results we use are from [LMS86 ]. We denote the suspension spectrum 1GX of a G*
*-space X simply
by X.
We recall the definition of homology and cohomology theories associated to a*
* G-equivariant
spectrum MG. Let X and Y be G-spectra. Let {X, Y }G denote the stable (weak) G*
*-homotopy
classes of maps from X to Y . We grade our theories by formal differences of G-*
*representations. For
brevity let ff denote the formal difference V - W of two finite dimensional rea*
*l G-representations
V and W . Let SffGdenote the spectrum S-W 1GSV . The homology is
MGff(X) = {SffG, MG ^ X}G ~={SVG, SWG^ MG ^ X}G.
The cohomology is
MffG(X) = {S-ffG^ X, MG}G ~={SWG^ X, SVG^ MG}G.
In this paper a ring spectrum E is a spectrum together with a multiplication*
* ~: E ^ E ! E
and a left unit j: S ! E for the multiplication in the stable homotopy category*
*. We do not need
to assume that E is associative nor commutative. An E-module spectrum M is a sp*
*ectrum with
an action E ^ M ! M by E that respects the unit and multiplication. Let EG be a*
* G-equivariant
ring spectrum. The coefficients EffG= EG-ffhave a bilinear multiplication that *
*is RO(G)-graded and
have a left unit element. Let MG be an EG-module spectrum. We have that M*G(X) *
*is naturally an
E*G-module, and MG*(X) is naturally an EG*-module for any G-spectrum X.
Let MG be a spectrum indexed on a G-universe U. For a closed subgroup H in G*
* let MH
denote MG regarded as an H-spectrum indexed on U now considered as a H-universe*
*. The forgetful
functor from G-spectra to H-spectra respects the smash product. A complete G-un*
*iverse U is also a
complete H-universe for all closed subgroups H of G. In lack of a reference we *
*include an argument
proving this well known result. Let V be a H-representation. The manifold GxV h*
*as a smooth and
free H-action given by h(g, v) = (gh-1, hv). It also has a smooth G-action by l*
*etting G act from
the right on G in G x V . The H-quotient G xH V is a smooth G-manifold [Bre72, *
*VI.2.5]. Now
consider the tangent G-representation W at (1, 0) 2 G xH V . This can be arrang*
*ed so that W is
an orthogonal G-representation by using a G-invariant Riemannian metric on G. C*
*ompactness of
H gives that V is a summand of W regarded as an H-representations.
5
Halvard Fausk
We have the following isomorphisms for any ff and any G-spectrum X [May96 , *
*XVI.4].
MHresG(X) ~={G=H+ ^ SffG, MG ^ X}G
Hff
Gff
MresHH(X) ~={G=H+ ^ S-ffG^ X, MG}G.
The forgetful functor from G-spectra to H-spectra respects the smash product.
We now consider induction and restriction maps. The collapse map c: G=H+ ! S*
*0 is the stable
map associated to the G-map that sends the disjoint basepoint + to the basepoin*
*t 0, and G=H to
1 in S0 = {0, 1}. Let o: S0 ! G=H+ be the transfer map [LMS86 , IV.2]. We recal*
*l a construction
of o after Proposition 3.1.
There is an induction map natural in G-spectra X and MG
indGH: MHresG(X) ! MGff(X).
Hff
It is defined by pre-composing with the transfer map S0 -o!G=H+ as follows
* G
MHresG(X) ~={G=H+ ^ SffG, MG ^ X}G -o!Mff(X).
Hff
There is a restriction map natural in G-spectra X and MG
resGH: MGff(X) ! MHresG(X).
Hff
It is defined by pre-composing with the collapse map G=H+ !cS0. The definition *
*is analogous for
cohomology. Alternatively, we can describe the induction map in cohomology as f*
*ollows:
MffG(o ^ 1X ): MffG(G=H+ ^ X) ! MffG(X)
and the restriction map as
MffG(c ^ 1X ): MffG(X) ! MffG(G=H+ ^ X)
composed with the isomorphism
(k^1X)* -ff resGHff
{S-ffG^ G=H+ ^ X, MG}G -! {G=H+ ^ SG ^ X, MG}G ~=MH (X)
where k: G=H+ ^ S-ff~=S-ff^ G=H+.
The classical Frobenius reciprocity law says that the induction map R(H) ! R*
*(G) between
representation rings is linear as an R(G)-module, where R(H) is given the R(G)-*
*module structure
via the restriction map. In our more general context the Frobenius reciprocity *
*law says that the
Gff
induction map MresHH(X) ! MffG(X) is linear as a map of E*G(S0)-modules (via th*
*e restriction
map). We need the following slightly different version.
Proposition 3.1. Let MG be a module over a ring spectrum EG. Let e 2 EHresHand *
*m 2 MGfi(X).
Gff
Then we have that
indGH(e) . m = indGH(e . resGHm)
in MGff+fi(X). The same result applies to cohomology.
Proof.Let e: G=H+ ^ SffG! EG represent the element e in EHresH, and let m: SfiG*
*! MG ^ X
Gff
represent the element m 2 MGfi(X).
We get that both products are
SffG^ SfiGo^1-!G=H+ ^ SffG^ SfiGe^m-!EG ^ MG ^ X
*
* __
composed with the EG ^ MG ! MG. The proof for cohomology is similar. *
* |__|
6
Generalized Artin and Brauer induction for compact Lie groups
We now describe the induction and restriction maps for homology theories in *
*more detail. This
is used in section 7. We have that
{Sff^ G=H+, E ^ X}G ~={Sff, D(G=H+) ^ E ^ X}G
where D(G=H+) is the Spanier-Whitehead dual of G=H+. Using the equivalences S-f*
*f^ G=H+ ~=
G=H+ ^ S-ffand D(G=H+) ^ E ~=E ^ D(G=H+) we get an isomorphism
EHresG(X) ~=EGff(D(G=H+) ^ X).
Hff
Under this isomorphism the induction map is given by EGff(D(o) ^ 1X ), and the *
*restriction map
as EGff(D(c) ^ 1X ). In the rest of this section we recall a description of the*
* transfer map [LMS86 ,
IV.2.3] and the Spanier-Whitehead dual of the collapse and transfer maps [LMS86*
* , IV.2.4]. Let M
be a smooth compact manifold without boundary. In our case M = G=H. There is an*
* embedding
of M into some finite dimensional real G-representation V [Bre72, VI.4.2]. The *
*normal bundle M
of M in V can be embedded into an open neighborhood_of M in V by the equivarian*
*t tubular
neighborhood theorem. The Thom construction ,of a bundle , on a compact manifol*
*d is equivalent
to the one point compactification of the bundle ,. We get a map
____
t0: SV ! M
by mapping everything outside of the tubular neighborhood of M to the point at *
*infinity.
*
*____
The Thom construction of the inclusion map__M ! M T M ~= V x M gives s0: *
* M !
SV ^ M+. Let the pretransfer_t: S0 ! S-V ^ M be S-V ^ t0pre-composed with S0 ~*
*=S-V ^ SV
and let s: S-V ^ M ! M+ be the composite of S-V ^ s0with S-V ^ SV ^ M+ ~=M+. T*
*he transfer
map o is defined to be the composite map
s O t: S0 ! M+.
We now let M be the G-manifold_G=H._Atiyah duality gives that the Spanier-White*
*head dual
of G=H+ is equivalent to S-V G=H . When G is finite this is just G=H+ itself. *
*The proof of the
equivariant Atiyah duality theorem [LMS86 , III.5.2] gives that D(c) ' t. It is*
* easy to see that
D(s) ' s. Hence we get
D(o) ' c O s and D(c) ' t.
The discussion above gives the following.
______
Lemma 3.2. Let c: G=H+ ! S0, s: S-V ^ G=H ! G=H+, and t: S0 ! S-V ^ G=H be as*
* above.
Then we have that the restriction map in homology is
______
(t ^ 1X )ff: EGff(X) ! EGff(S-V G=H ^ X)
and the induction map is
______
((c O s) ^ 1X )ff: EGff(S-V G=H ^ X) ! EGff(X)
______
composed with the isomorphism EGff(S-V G=H ^ X) ~=EHresH(X).
Gff
If G is a finite group, then s is the identity map. Hence the induction map *
*is the induced map
from the collapse map c, and the restriction map is the induced map from the tr*
*ansfer map o,
composed with the isomorphism EGff(G=H+ ^ X) ~=EHresH(X).
Gff
4. The Burnside ring
The stable homotopy classes of maps between two G-spectra are naturally modules*
* over the Burnside
ring of G. We use this Burnside ring module structure to prove our induction th*
*eorems.
7
Halvard Fausk
We recall the following description of the Burnside ring A(G) of a compact L*
*ie group G from
[tD75, tD79, LMS86]. Let a(G) be the semiring of isomorphism classes of compact*
* G-CW-complexes
with disjoint union as sum, cartesian product as product, and the point as the *
*multiplicative unit
object. Let C( G; Z), or C(G) for short, be the ring of continuous functions fr*
*om the space G of
conjugacy classes of closed subgroups of G with finite Weyl group to the intege*
*rs Z. Let OEu(X)
denote the Euler characteristic of a space X. We define a semi-ring homomorphis*
*m from a(G)
to C(G) by sending X to the function (H) 7! OEu(XH ) [tD79, 5.6.4]. This map ex*
*tends to a
ring homomorphism OE0from the Grothendieck construction b(G) of a(G) to C(G). T*
*he Burnside
ring A(G) is defined as b(G)= kerOE0. We get an injective ring map OE: A(G) ! C*
*(G). The image
of OE is generated by OE(G=H+) for H 2 G. One can show that C(G) is freely gen*
*erated by
|WGH|-1OE(G=H+) for H 2 G [LMS86 , V.2.11]. We have that |G|C(G) A(G) where *
*|G| is the
order of G [tD77, thm.2]. We denote the class O(X) in A(G) corresponding to a f*
*inite G-CW-complex
X by [X]. Define a ring homomorphism d: ssG0(S0G) ! C(G) by sending a stable ma*
*p f: S0G! S0G
to the function that sends (H) in G to the degree of the fixed point map fH . *
*It follows from
Montgomery and Zippin's theorem that the maps OE0and d take values in continuou*
*s functions from
G to Z.
There is a map O: A(G) ! ssG0(S0) given by sending the class of a compact G-*
*CW-complex X
to the composite of the transfer and the collapse map
S0G! X+ ! S0G.
The map O is the categorical Euler characteristic [LMS86 , V.1]. It turns out t*
*o be a ring homo-
morphism [LMS86 , V.1]. It has the property that the degree of the H-fixed poin*
*t of a map in the
homotopy class O(X) is equal to OEu(XH ) (the ordinary Euler characteristic of *
*the fixed point space
XH ) [LMS86 , V.1.7]. A proof is given in Lemma 4.2. We have the following comm*
*utative triangle
O
A(G) ________________//_ssG0(S0G).
KK rr
KKK rrr
OEKKK%%KKxxrrrdrr
C( G; Z)
A theorem, due to Segal when G is a finite group and to tom Dieck when G is a c*
*ompact Lie group,
says the map O is an isomorphism [LMS86 , V.2.11]. This alow us to use the foll*
*owing three different
descriptions of elements in the Burnside ring:
i)Formal differences of equivalence classes of compact G-CW complexes
ii)Stable homotopy classes of self maps of S0G
iii)Certain continuous functions from G to the integers.
Since the Burnside ring A(G) is isomorphic to {S0G, S0G}G we have that G-equ*
*ivariant cohomology
and homology theories naturally take values in the category of modules over A(G*
*).
In the rest of this section we prove that the degree of the H-fixed points o*
*f a stable map
f: S0G! S0Gis the same as the degree of the !(H)-fixed point of f for any close*
*d subgroup H of
G. The following is well known.
Lemma 4.1. Let X be any space with an action by a torus T . If both OEu(X) and *
*OEu(XT ) exist,
then OEu(X) = OEu(XT ).
Proof.Replace X by a weakly equivalent T -CW-complex. We get that the quotient *
*complex X=XT
is built out of one single point * and cells Dn ^ T=A for proper subgroup A of *
*T . All nontrivial
cosets of T are tori (of positive dimension). Hence all the cells have Euler ch*
*aracteristic equal to 0
except for the point which has Euler characteristic 1. The claim follows by the*
* long exact_sequence
in homology and the assumptions that both OEu(X) and OEu(XT ) exist. *
* |__|
8
Generalized Artin and Brauer induction for compact Lie groups
The following is a generalization of [LMS86 , V.1.7]. They consider closed s*
*ubgroups with finite
Weyl group.
Lemma 4.2. Let X be a compact G-CW-complex and let f: S0G! S0Gbe a stable G-map*
* in the
stable homotopy class O(X) 2 ssG0(S0G). Then deg(fL ) = OEu(XL) for any closed *
*subgroup L of G.
Proof.The geometric fixed point functor L is a strong monoidal functor from th*
*e stable homotopy
category of G-spectra to the stable homotopy category of WGL-spectra [LMS86 , I*
*I.9.12]. We also
have that L( 1GX+) ~= 1WGLXL+[May96 , XVI.6]. The forgetful functor from the s*
*table WGH-
homotopy category to the nonequivariant stable homotopy category is also strong*
* monoidal. The
categorical Euler characteristic respects strong monoidal functors [May01 , 3.2*
*]. Hence we get that
deg(fL ) is equal to the degree of the categorical Euler characteristic of the *
*spectrum _L(_1GX)
regarded as a nonequivariant spectrum. This is OEu(XL). *
* |__|
Proposition 4.3. Let f: S0G! S0Gbe a stable G-map. Let H be a closed subgroup o*
*f G and let
!(H) be the associated conjugacy class of subgroups with finite Weyl group. The*
*n we have that
deg(fH ) = deg(f!(H)).
Proof.Let K be a subgroup in the conjugacy class !(H) so that H < K and K=H is *
*a torus. Let
X and Y be finite G-CW-complexes such that f is in the homotopy class O(X) - O(*
*Y ). Since
OEu(XK ) = OEu((XH )K=H ) and OEu(Y K) = OEu((Y H)K=H )
the previous two Lemmas give that
deg(fK ) = OEu(XK ) - OEu(Y K) = OEu(XH ) - OEu(Y H) = deg(fH ).
*
* __
*
*|__|
We need the following corollary in section 7.
Corollary 4.4. Let f: S0G! S0Gbe a map such that deg(fA ) = 0 for all abelian s*
*ubgroups of G
with finite Weyl group. Then f is null homotopic restricted to the K-equivarian*
*t stable homotopy
category for any abelian subgroup K of G.
Proof.Proposition 4.3 together with Lemma 2.3 give that the degree deg(fK ) = 0*
* for all abelian
subgroups A of G. The claim follows since a self map of S0Kis null homotopic if*
* and only if all_the
degrees of all its fixed point maps are 0 [tD79, 8.4.1]. *
* |__|
5.Artin induction
We first introduce some conditions on ring spectra and then we prove the Artin *
*induction and
restriction theorems. Recall definition 2.4. Let Jn be the A(G)-ideal consistin*
*g of all elements fi 2
A(G) such that deg(fiA ) = 0 for all (A) 2 An. Let J be the intersection of all*
* Jn.
Definition 5.1. We say that a G-equivariant ring spectrum EG satisfies the n-in*
*duction property
if JnE0G= 0. We say that EG satisfies the induction property if JE0G= 0.
Let j: S0 ! EG be the unit map of the ring spectrum EG. We have that EG sati*
*sfies the
n-induction property if and only if the ideal Jn is in the kernel of the unit m*
*ap
j: A(G) ! E0G.
If EG satisfies the n-induction property and E0Gis an EG-algebra, then E0Galso *
*satisfies the n-
induction property.
9
Halvard Fausk
Let eH : G ! Z be the function defined by letting eH (H) = 1 and eH (K) = 0*
* for (K) 6= (H).
We have that eA is a continuous function for every (A) 2 A since (A) is an open*
*-closed point in G
by Lemma 2.1. Since |G|C(G) OEA(G) we have that |G|eA 2 OEA(G) for all A 2 A.*
* When G is a
compact Lie group it turns out that we can sharpen this result. Recall Definiti*
*on 2.8.
Proposition 5.2. Let (K) be an element in An. Then |G|neK is an element in OEA(*
*G). Moreover,
the element can be written as
P
|G|neK = cAOE(G=A)
(A)6(K)
where cA 2 Z and (A) 2 An.
Proof.Let S(K) denote the subset of G consisting of all (A) 6 (K) in G. Lemma*
* 2.1 and 2.7
imply that if K is in An, then S(K) is a finite subset of An consisting of open*
*-closed points in
G. We prove the Proposition by induction on the length of chains (totally orde*
*red subsets) in the
partially ordered set S(K). If (K) is minimal in An, then OE(G=K) = |WGK|eK and*
* the claim is
true. Assume the claim is true for all (A) such that all chains in S(A) have le*
*ngth (l - 1) or less. If
all chains of subgroups in S(K) have length l or less, then we get
__|G|n_ P
OE(G=K) = |G|neK + |G|nmAeA
|WGK| (A) (K)
where mA = |(G=K)A|=|WGK| are integers (as explained in the beginningPof sectio*
*n 2). By the
inductive hypothesis we get that |G|neK is in OEA(G), and |G|neK = cAOE(G=A) *
*where_the sum is
over (A) 6 (K). *
* |__|
WePconclude that there is a stable map ffn: S0G! S0Gwhose degree function d(*
*ffn) 2 C(G)
is (A)2An|G|neGA. The degree of (|G|n - ffn)A is 0 for all (A) 2 An. Hence if*
* EG satisfies the
n-induction property, then (|G|n - ffn)E*G= 0. So |G|nE*Gis equal to ffnE*G.
The element [G=H] in A(G) corresponds, via the isomorphism O, to
S0G-o!G=H+ -c!S0G
in ssG0(S0G). Here o is the transfer map and c is the collapse map. Hence we ha*
*ve that
[G=H] = indGH[*].
The isomorphism class of a point [*] 2 A(H) correspond to the identity map in s*
*sH0(S0H).
The next result is the Artin induction theorem.
Theorem 5.3. Assume that EG is a ring spectrum satisfying the n-induction prope*
*rty. Then the
integer |G|n times the unit element in E*Gis in the image of the induction map
(A)2AnindGA: (A)2AnE0A! E0G
where the sum is over representatives for each conjugacy class (A) 2 An.
Proof.We have that |G|n1 = ffn1 in EffG. The Proposition follows from the Frobe*
*nius reciprocity law
3.1 and Lemma 5.2. More precisely, let f: S0G! EG represent an element in E0G. *
*Then [G=H] ._f =
indGH[c ^ f], where c ^ f: G=H+ ! EG represents an element in E0H~=E0G(G=H+). *
* |__|
As a consequence of the Artin induction theorem we can reconstruct EG(X), ra*
*tionally, from all
the EA(X) with (A) 2 An and the restriction and conjugation maps. To do this we*
* need the double
coset formula for compact Lie groups. The double coset formula was first proved*
* by M. Feshbach
[Fes79]. We follow the presentation given in [LMS86 , IV.6]. To state the doubl*
*e coset formula it is
convenient to express the restriction and induction maps between EH and EK for *
*subgroups H and
10
Generalized Artin and Brauer induction for compact Lie groups
K of G by maps in the G-stable homotopy category. Let H 6 K be subgroups of G. *
*There is a
collapse map
cKH: G=H+ ! G=K+
and a transfer map
oKH: G=K+ ! G=H+,
which induce restriction and induction maps [LMS86 , p.204]. Let g be an elemen*
*t in G. Right
multiplication by g induces an equivalence of G-manifolds
fig: G=H+ ! G=(g-1Hg)+.
Consider G=H as a left K-space. The space is a compact differentiable K-manifol*
*d so it has finitely
many orbit types [tD79, 5.9.1]. The orbit type of an element x is the K-isomorp*
*hism class of the
homogeneous space Kx. The stabilizer of the element gH is K \ gHg-1. The left K*
*-quotients of
subspaces of G=H consisting of all points of a fixed orbit type are manifolds [*
*Bre72, IV.3.3]. These
manifolds are called the orbit type manifolds of K\G=H. We decompose the double*
* coset space
K\G=H as a disjoint union of the connected components Miof all the orbit type m*
*anifolds. We are
now ready to state the double coset formula.
Theorem 5.4. Let G be a compact Lie group and H and K be closed subgroups of G.*
* Then we
have
P gHg-1
oGHO cGK' Miz(Mi)figO cK\gHg-1O oKK\gHg-1
where the sum is over orbit-type manifold components Miand g 2 G is a represent*
*ative of each Mi.
The integer z(Mi) is the internal Euler characteristic. It is the Euler charact*
*eristic of the closure of
Miin K\G=H minus the Euler characteristic of its boundary.
Note that the transfer
oKH: G=K+ ! G=H+
is trivial if the Weyl group WK H is infinite. In particular, we have the follo*
*wing [Fes79, II.17]
[LMS86 , IV.6.7].
Lemma 5.5. Assume H = K is a maximal torus T in G. Then in the double coset for*
*mula it is
enough to take the sum over elements g 2 G representing each gT in the Weyl gro*
*up WGT of T .
Let EG be a G-equivariant ring spectrum that satisfy the n-induction propert*
*y, and let MG be
a module over EG. There is a restriction map
Q Q
MffG(X) ! Eq[ AMresffA(X) ' K,L,gMresffK\gLg-1(X)].
The first product in the equalizer is over representatives for conjugacy classe*
*s of An, and the second
product is over pairs K, L of these subgroup representatives and over represent*
*atives g for each of
the orbit-type manifold components of K\G=L. The maps in the equalizer are the *
*two restriction
(and conjugation) maps.
Definition 5.6. Let r be an integer. We say that a pair of maps
f: A ! B and g: B ! A
between abelian groups is an r-isomorphism pair if f Og = r and gOf = r. A map *
*is an r-isomorphism
if it is a map belonging to an r-isomorphism pair.
The Artin induction theorem implies the following Artin restriction theorem.
11
Halvard Fausk
Theorem 5.7. Let EG be a G-ring spectrum satisfying the n-induction property. L*
*et MG be an
EG-module spectrum. Then there exists a homomorphism
Q Q
_: Eq[ AMresffA(X) ' K,L,gMresffK\gLg-1(X)] ! MffG(X)
such that the restriction map and _ is a |G|n-isomorphism pair. The first produ*
*ct in the equalizer is
over representatives for conjugacy classes of An, and the second product is ove*
*r pairs K, L of these
subgroup representatives and over representatives g for each of the orbit-type *
*manifold components
of K\G=L.
There is a similar result for homology.
Proof.The following argument is standard. Our proof is close to [McC86 , 2.1]. *
*We prove the result
in the following generality. Consider an element r 2 E0Gin the image of the ind*
*uction maps from
E0Hifor a set of subgroups Hiof G. In our case r = |G|n and the subgroups are r*
*epresentatives for
the conjugacy classes An by the Artin induction Theorem 5.3.
P k
Let r = i=1indGHiriwhere ri2 EHi. Define _ by setting
Q P k
_( HimHi) = i=1indGHi(rimHi).
We have that
P k P k
_ O res(m) = i=1indGHi(riresGHim) = i=1indGHi(ri)m = rm.
The second equality follows from Frobenius reciprocity law 3.1.
Q
We now consider the projection of resO _( mHi) to MK . It is
P G G
iresKindHi(rimHi).
By the double coset formula 5.4 this equals
P P K gHig-1
i KgHiziindgHig-1\KresgHig-1\Kfig(rimHi)
where for each i the sum is over representatives KgHiof components of orbit-typ*
*e manifolds of the
double coset K\G=Hiand ziis an integer. By our assumptions we have that
-1 K
resgHiggHig-1\KmgHig-1= resgHig-1\KmK .
So by Frobenius reciprocity we get
P iP K gHig-1 j
i KgHiziindgHig-1\KresgHig-1\KfigrimK .
This equals
P G G G
i(resKindHi(ri))mK = resK(r)mK . *
* __
*
*|__|
For a fixed Lie group G both the restriction map and the map are natural i*
*n MG and X.
6.Brauer induction
We present an integral induction theorem for cohomology theories satisfying the*
* n-induction prop-
erty. We first discuss some classes of subgroups.
Definition 6.1. A subgroup H of G is n-hyper if it has finite Weyl group and th*
*ere is an extension
0 ! A ! H ! P ! 1
such that:
12
Generalized Artin and Brauer induction for compact Lie groups
i)P is a finite p-group for some prime number p
ii)A is an abelian subgroup of G, such that !(A) is topologically generated by*
* n or fewer elements,
and |A=AO| is relatively prime to p.
Lemma 6.2. Let H be an n-hyper subgroup of G (for the prime p) and let K be a s*
*ubgroup of H.
Then K is an n-hyper subgroup of G (for the prime p) if K has finite Weyl group*
* in G.
*
* __
Proof.This follows from Lemma 2.7. *
* |__|
We next describe the idempotent elements in the Burnside ring A(G) localized*
* at a rational
prime. First we need some definitions. A group H is said to be p-perfect if it *
*does not have a
nontrivial (finite) quotient p-group. The maximal p-perfect subgroup H0pof H is*
* the preimage in H
of the maximal p-perfect subgroup of the group of components H=HO. Let H be a s*
*ubgroup of a
fixed compact Lie group G. Let Hp denote the conjugacy class !(H0p) of subgroup*
*s of G with finite
Weyl group associated to H0p. Let pG denote the subspace of G consisting of c*
*onjugacy classes
of all p-perfect subgroups of G with finite Weyl group in G. Let m(p)denote the*
* largest factor of
an integer m that is relatively prime to p. The next result is proved for finit*
*e groups in [tD79, 7.8],
and for compact Lie groups in [FO05 , 3.4].
Theorem 6.3. Let G be a compact Lie group. Let H be a p-perfect subgroup such t*
*hat (H) 2 pG
is an open-closed point in pG. Let m be an integer such that m eH is in OEA(G)*
*. Then there exists
an idempotent element IH,p2 C( G, Z) such that m(p)IH,p2 OEA(G), and IH,pevalua*
*ted at (K) is
1 if Kp = (H) and zero otherwise. In particular, IH,pis an idempotent element i*
*n the localized ring
OEA(G)(p).
An abelian group A is p-perfect if and only if |A=AO| is relatively prime to*
* p by Lemma 2.6.
We can apply Theorem 6.3 to p-perfect abelianPgroups in An with m = |G|n by Lem*
*ma 2.1 and
Proposition 5.2. Let I(p,n)be (|G|n)(p) (A)IA,pwhere the sum is over all (A) 2 *
*An such that |A=AO|
is relatively prime to p. The element I(p,n)2 C(G) is in OEA(G). The function I*
*(p,n)has the value
(|G|n)(p)at each conjugacy class (H) of the form
0 ! S ! H ! P ! 1
where P is a p-group and S is abelian with |S=SO| relatively prime to p and !(S*
*) 2 An, and I(p,n)
has the value 0 at all other elements of G. In particular, I(p,n)has the value*
* (|G|n)(p)at each
A 2 An by Lemma 2.6. The greatest common divisor of {(|G|n)(p)}, wherePp runs o*
*ver primes p
dividing |G|n,Pis 1. Hence there is a set of integers zp such that p||G|zp|G|*
*(p)= 1. Let In be
the element pzpI(p,n)where the sum is over primes p dividing |G|. The element*
* In is not an
idempotent element in C(G). The function In: G ! Z has the value 1 for all (A)*
* 2 An. Since In is
in the image of OE: A(G) ! C(G), there is a map fin: S0G! S0Gso that the degree*
* d(fin) is In. The
degree of (1 - fin)A is zero for all (A) 2 An. Assume EG is a G-equivariant rin*
*g spectrum satisfying
the n-induction property. We get that
E*G(S0) = fin . E*G(S0).
Lemma 6.4. The element In 2 A(G) can be written as
P
In = OE( kHi[G=Hi])
(Hi)
where the subgroups Hiare n-hyper subgroups of G for primes dividing |G|n or su*
*bgroups of such,
and kHi are integers.
Proof.We know that In can be written in the above form for some Hi. Let Hj be a*
* maximal
subgroup of G in the sum describing In. The value of In at (Hj) is kHj|WGHj| 6=*
* 0. Hence the_
maximal subgroups are n-hyper. So the Hiare n-hyper subgroups or subgroups of s*
*uch. |__|
13
Halvard Fausk
The next result is the Brauer induction theorem.
Theorem 6.5. Assume EG is a ring spectrum satisfying the n-induction property. *
*Then the unit
element 1 in E*Gis in the image of the induction map
(H)indGH(X): (H)E0H! E0G
where the sum is over n-hyper subgroups of G for primes p dividing |G|n.
Proof.We get as in Theorem 5.3 that the unit element 1 is in the image of the i*
*nduction maps from
all the subgroups Hiin Lemma 6.4. The conclusion follows by noting that for sub*
*groups K 6_H_in
G we have indGKx = indGH(indHKx) by [LMS86 , IV.7.1]. *
* |__|
As a consequence we get the following Brauer restriction theorem.
Theorem 6.6. Let EG be a G-equivariant ring spectrum satisfying the n-induction*
* property, and
let MG be a module over EG. Then the restriction map
Q Q
MffG(X) ! Eq[ H MresffH(X) ' K,L,gMresffK\gLg-1(X)]
is an isomorphism. The first product in the equalizer is over representatives f*
*or conjugacy classes of
n-hyper subgroups of G for primes dividing |G|n, and the second product is over*
* pairs K, L of these
subgroup representatives and over representatives g for each of the orbit-type *
*manifold components
of K\G=L.
*
* __
Proof.This follows from the proof of Theorem 5.7 and the Brauer induction theor*
*em 6.5. |__|
There is a similar result for homology.
7.Induction theory for Borel cohomology
Let k be a nonequivariant spectrum. The Borel cohomology and Borel homology on *
*the category of
based G-spaces are
k*(X ^G EG+) andk*( Ad(G)X ^G EG+).
The adjoint representation Ad(G) of G is the tangent vectorspace at the unit el*
*ement of G with
G-action induced by the conjugation action by G on itself. If G is a finite gro*
*up, then Ad(G) = 0.
Borel homology and cohomology can be extended to an RO(G)-graded cohomology the*
*ory defined
on the stable equivariant homotopy category. We follow Greenlees and May [GM95 *
*].
Let MG be a G-spectrum. The geometric completion of MG is defined to be
c(MG) = F (EG+, MG)
where F denotes the internal hom functor in the G-equivariant stable homotopy c*
*ategory. Let
f(MG) be MG ^ EG+. The Tate spectrum of MG is defined to be
t(MG) = F (EG+, MG) ^ "EG.
The space "EG is the cofiber of the collapse map EG+ ! S0.
If EG is a ring spectrum, then c(EG) and t(EG) are also ring spectra [GM95 ,*
* 3.5]. More precisely,
c(EG) is an algebra over EG, and t(EG) is an algebra over c(EG). The product on*
* the spectrum
f(EG) is not unital in general, however it is a c(EG) module spectrum. Hence we*
* have the following.
Proposition 7.1. Let EG be a G-ring spectrum satisfying the n-induction propert*
*y. Then c(EG)
and t(EG) are G-equivariant ring spectra satisfying the n-induction property. T*
*he spectrum f(EG)
is a c(EG)-module spectrum.
14
Generalized Artin and Brauer induction for compact Lie groups
Let i: UG ! U be the inclusion of the universe UG into a complete G-universe*
* U. Let i*k denote
the G-spectrum obtained by building in suspensions by G-representations. If k i*
*s a ring spectrum,
then i*k is a G-equivariant ring spectrum. The following results is proved in [*
*GM95 , 2.1,3.7].
Proposition 7.2. Let k be a spectrum. Let X be a naive G-spectrum (indexed on U*
*G ). Then we
have isomorphisms
(c(i*k))nG(X) ~=kn(X ^G EG+) and(f(i*k))Gn(X) ~=kn( Ad(G)X ^G EG+)
where Ad(G) is the adjoint representation of G.
We next show that c(k) has the induction property (see Definition 5.1) when *
*k is a complex
oriented spectrum. For every compact Lie group G, there is a finite dimensional*
* unitary faithful
G-representation V . Hence G is a subgroup of the unitary group U(V ). Let the *
*flag manifold F
associated to the G-representation V be U(V )=T where T is some fixed maximal t*
*orus of U(V ).
Let G act on F via the embedding G 6 U(V ). The following result is well known *
*[HKR00 , 2.6].
Proposition 7.3. Let k be a complex oriented spectrum. Then the map
k*(BG+) ! k*(F+ ^G EG+),
induced by the collapse map F ! *, is injective.
Theorem 7.4. Given a compact Lie group G let N be an integer so that AN = A. Le*
*t k be a complex
oriented spectrum. Then there exists an integer d so that the Borel cohomology *
*G-spectrum c(i*k)
satisfies the following (|G|N - ffN )dc(i*k)* = 0 and (1 - fiN )dc(i*k)* = 0.
The class ffN is defined after Proposition 5.2 and the class fiN is defined *
*before Lemma 6.4.
Proof.Easy induction gives the following: Let Y be a d-dimensional G-CW complex*
*, and let EG
be a cohomology theory. Assume that an element r 2 A(G) kills the E*G-cohomolog*
*y of all the cells
in Y . Then rd kills E*G(Y ).
By Proposition 7.3 it suffices to show that rd(i*k)*(U=T ) = 0 when r is |G|*
*N - ffN or 1 - fiN .
We have that U=T is a finite G-CW complex with orbit types G=(G \ gT g-1) for g*
* 2 G. Since both
|G|N - ffN and 1 - fiN restricted to any abelian group is 0 by Corollary 4.4, w*
*e get that they kill
the c(i*k)-cohomology of all the cells in Y = U=T . We can take d to be the dim*
*ension_of the G-CW
complex U=T . *
* |__|
The Propositions 7.3 and 7.4 give Artin and Brauer restriction theorems for *
*c(k) and t(k) where
k is a complex oriented spectrum. We state the theorems only for c(k) applied t*
*o naive G-spectra.
Theorem 7.5. Let k be a complex oriented cohomology theory. Then for any naive *
*G-spectrum X
the restriction map from k*(X ^G EG+) to the equalizer of
Q * Q *
Ak (X ^A EG+) ' K,L,gk (X ^K\gLg-1EG+)
is a natural isomorphism after inverting |G|. The first product is over represe*
*ntatives for conjugacy
classes of A, and the second product is over pairs K, L of these subgroup repre*
*sentatives and over
representatives g for each of the orbit-type manifold components of K\G=L.
Theorem 7.6. Let k be a complex oriented cohomology theory. Then for any naive *
*G-spectrum X
the restriction map from k*(X ^G EG+) to the equalizer of
Q * Q *
Ak (X ^A EG+) ' K,L,gk (X ^K\gLg-1EG+)
is a natural isomorphism. The first product is over representatives for conjuga*
*cy classes of hyper
subgroups of G for primes dividing |G|, and the second product is over pairs K,*
* L of these subgroup
representatives and over representatives g for each of the orbit-type manifold *
*components of K\G=L.
15
Halvard Fausk
Remark 7.7. See also [Fes81] and [LMS86 , IV.6.10].
Since f(i*k) is a c(i*k) module spectrum we get Artin and Brauer restriction*
* theorems for the
Borel homology k*(EG ^G Ad(G)X) when k is complex oriented.
Theorem 7.8. Let k be a complex oriented cohomology theory. Then for any naive *
*G-spectrum X
the restriction map from k*( Ad(G)X ^G EG+) to the equalizer of
Q Ad(A) Q Ad(K\gLg-1)
Ak*( X ^A EG+) ' K,L,gk*( X ^K\gLg-1EG+)
is a natural isomorphism after inverting |G|. The first product in the equalize*
*r is over representatives
for conjugacy classes of A, and the second product is over pairs K, L of these *
*subgroup representa-
tives and over representatives g for each of the orbit-type manifold components*
* of K\G=L.
Theorem 7.9. Let k be a complex oriented cohomology theory. Then for any naive *
*G-spectrum X
the restriction map from k*( Ad(G)X ^G EG+) to the equalizer of
Q Ad(A) Q Ad(K\gLg-1)
Ak*( X ^A EG+) ' K,L,gk*( X ^K\gLg-1EG+)
is a natural isomorphism. The first product is over representatives for conjuga*
*cy classes of hyper
subgroups of G for primes dividing |G|, and the second product is over pairs K,*
* L of these subgroup
representatives and over representatives g for each of the orbit-type manifold *
*components of K\G=L.
It is immediate from the definition that the induction and restriction maps *
*in Borel cohomology
are given by the transfer and the collapse maps
o: S0 ! G=H+ and c: G=H+ ! S0
as follows
* *
k*(X ^H EG+) ~=k*((G=H+ ^ X) ^G EG+) o!k (X ^G EG+)
and
k*(X ^G EG+) c*!k*((G=H+ ^ X) ^G EG+) ~=k*(X ^H EG+).
When G is a finite group the induction map in Borel homology is induced from*
* the collapse
map c and the restriction map is induced from the transfer_map o. This is more *
*complicated for
compact Lie groups. The Spanier Whitehead dual S-V G=H of G=H+ (in the stateme*
*nt of Lemma
3.2) is equivalent to G nH S-L(H) where n is the halfsmash product [May96 , XVI*
*.4] and L(H) is
the H-representation on the tangent space at eH in G=H induced by the action h,*
* gH 7! hgH. By
considering
H ! G ! G=H
we see that Ad(G) is isomorphic to Ad(H) L(H) as H-representations. By proper*
*ties of the half
smash product we have that
i j
G nH S-L(H) ^ SAd(G)~=G nH SAd(G)-L(H)~=G nH SAd(H)
and
(G nH SAd(H)^ X) ^G EG+ ~=(SAd(H)^ X) ^H EG+.
Combined with 3.2 this give a description of induction and restriction maps.
Let G be a finite group. We state some result from [HKR00 ] to show that und*
*er some hypothesis
a local complex oriented cohomology theory satisfies n-induction. Let k* be a l*
*ocal and complete
graded ring with residue field of characteristic p > 0. Assume that
k0(BG) ! p-1k0(BG)
is injective. Assume that the formal group law of k* modulo the maximal ideal h*
*as height n.
16
Generalized Artin and Brauer induction for compact Lie groups
In [HKR00 ] the authors show that the restriction maps from
p-1k*(BG)
into the product of all p-1k*(BA) for all p-groups (A) 2 An is injective.
Hence if fi is an element in the Burnside ring A(G) of G such that deg(fiA )*
* = 0 for all A in An,
then we have that
fik*(BG) = 0.
So the Borel cohomology k*(X ^G EG+) satisfies the n-induction property. The au*
*thors also show
that k*(X ^G EG+) does not satisfy the n - 1 induction property.
8.0-induction, singular Borel cohomology
We consider induction theorems for ordinary singular Borel cohomology. Let M be*
* a G=GO-module.
The Borel cohomology of a unbased G-space X with coefficients in M is singular *
*cohomology of the
Borel construction of X
H*(X xG EG; M)
with local coefficients via
(X xG EG) ! (* xG EG) ! ss1(* xG EG) ~=G=GO
where is the fundamental groupoid. Since BG is path connected the fundamental*
* groupoid of BG
is non-canonically isomorphic to the one-object category ss1(BG), which is the *
*group of components
of G. Borel cohomology is an equivariant cohomology theory.
Lemma 8.1. The Borel cohomology of a unbased G-space X with coefficients in a G*
*=GO-module
M is represented in the stable G-equivariant homotopy category by the geometric*
* completion of an
Eilenberg-Mac Lane spectrum HM", where M"is a Mackey functor so that M"(G=1) is*
* isomorphic to
M as a G-module.
Pre-composing with the functor X 7! 1GX+ is implicit in the statement that *
*the Borel coho-
mology is represented.
Proof.The Borel cohomology is isomorphic to the cohomology of the cochain compl*
*ex of G-
homomorphisms from the G-cellular complex of X x EG to the G-module M [Hat02, 3*
*.H]. Assume
that M" is a Mackey functor so that M"(G=1) is isomorphic to M as a G-module. B*
*y the cell com-
plex description of Bredon cohomology we get that the Borel cohomology group is*
* isomorphic to
the G-Bredon cohomology of X x EG with coefficients in "M. It follows from [GM9*
*5 , 6.1]_that_there
are Mackey functors of the requested form. *
* |__|
The relation between the geometric completion of Eilenberg-Mac Lane spectra *
*and classical Borel
cohomology theory is also treated in [GM95 , 6,7]. From now on we consider the *
*Borel cohomology
defined on G-spectra.
The restriction map on the zeroth coefficient groups of the geometric comple*
*tion of HM is
described by the following commutative diagram
~=
H0(BG+; M) ____//_MG
res|| ||
fflffl|~= fflffl|
H0(B*+; M) ____//_M.
17
Halvard Fausk
We have that res(fi)res(m) = deg(fi)res(m) for any fi 2 A(G) and m 2 MG . Since*
* the restriction
map is injective we get that fim = deg(fi)m. So if deg(fi) = 0, then fiH0(BG+; *
*M) = 0. We have
that deg(fi) = deg(fiT) for any torus T in G by Proposition 4.3. So singular Bo*
*rel cohomology
satisfies the 0-induction property. The argument above applies more generally t*
*o show that Bredon
homology and cohomology with Mackey functor coefficients M such that M(G=G) ! M*
*(G=e) is
injective (or, alternatively, M(G=e) ! M(G=G) is surjective) satisfies 0-induct*
*ion.
The Artin restriction theorem gives a refinement of Borel's description of r*
*ational Borel coho-
mology. See also [Fes81, II.3]. Recall Lemma 5.5.
Theorem 8.2. Let G be a compact Lie group, X a G-spectrum, and M a G=GO-module.*
* Then the
restriction map
H*(X ^G EG+; M) ! H*(X ^T EG+; M)WGT
is a |WGT |-isomorphism.
In particular, with X = S0 we get that
H*(BG+; M) ! H*(BT+; M)WGT
is a |WGT |-isomorphism.
We next give the Brauer restriction theorem. Recall Lemma 5.5.
Theorem 8.3. Let G be a compact Lie group, X a G-spectrum and M a G=GO-module. *
*Fix a
maximal torus T in G. Then the restriction map
H*(X ^G EG+; M) ! limH*(X ^K EG+; M)
K
is an isomorphism. The limit is over all the subgroups K of G that have a norma*
*l abelian subgroup
A of K so that A 6 T and K=A is a p-group for some prime p dividing |WGT |. The*
* maps in the
limit are restriction maps an conjugation maps.
We use singular homology with local coefficients.
Theorem 8.4. Let G be a compact Lie group, X a G-spectrum, and M a G=GO-module.*
* Then the
restriction map (induced by t)
H*(SAd(G)X ^G EG+; M) ! H*(SAd(T)X ^T EG+; M)WGT
is a |WGT |-isomorphism.
There is also a Brauer restriction theorem for homology
Theorem 8.5. Let G be a compact Lie group, X a G-spectrum and M a G=GO-module. *
*Fix a
maximal torus T in G. Then the restriction map
H*(SAd(G)X ^G EG+; M) ! limH*(SAd(K)X ^K EG+; M)
K
is an isomorphism. The limit is over all the subgroups K of G that have a norma*
*l abelian subgroup
A of K so that A 6 T and K=A is a p-group for some prime p dividing |WGT |. The*
* maps in the
limit are restriction maps an conjugation maps.
9.1-induction, K-theory
In this section we consider equivariant K-theory KG. For details on KG see [Seg*
*68b]. An element
g 2 G is said to be regular if the closure of the cyclic subgroup generated by *
*g has finite Weyl group
in G.
18
Generalized Artin and Brauer induction for compact Lie groups
Definition 9.1. Let aeG denote the space of conjugacy classes of regular elemen*
*ts in G. Define
r: aeG ! G by sending a regular element g in G to the closure of the cyclic su*
*bgroup generated
by g.
The space aeG is given the quotient topology from the subspace of regular el*
*ements in G. The
map r is continuous since two regular elements in the same component of G gener*
*ate conjugate
cyclic subgroups [Seg68a, 1.3]. Let C(X, R) denote the ring of continuous funct*
*ions from a space
X into a topological ring R. The map r, together with the inclusion of Z in C, *
*induces a ring
homomorphism
r*: C( G, Z) ! C(aeG, C).
The ring of class functions on G is a subring of C(aeG, C) since the regular el*
*ements of G are dense
in G. Let R(G) denote the (complex) representation ring of G. Let O: R(G) ! C(a*
*eG, C) be the
character map. The map O is an injective ring map. Let V be a G-representation.*
* The value of O(V )
at (a regular element) g 2 G is the trace of g: V ! V . This only depends on th*
*e isomorphism class
of V .
We now give a detailed description of the induction map indGH: R(H) ! R(G) f*
*or the coefficient
ring of equivariant K-theory [Oli98, Seg68a]. Let , be a H-character. On a regu*
*lar element g in G
the induction map is given by
P
indGH,(g) = ,(k-1gk)
kH
where the sum is over the finite fixed set (G=H)g. If x 2 R(H), then indGH,(x) *
*is in the image of
,: R(G) ! C(aeG, C) [Oli98, 2.5]. So we get a well defined map indGH: R(H) ! R(*
*G). This is the
induction map defined in section 3 by [Nis78, 5.2] and [Seg68a, 2].
The unit map in G-equivariant K-theory induces a map oe: A(G) ! R(G). It is *
*the generalized
permutation representation map
P
oe([X]) = (-1)i[Hi(X; C)]
i
where [Hi(X; C)] is the isomorphism class of the G-representation Hi(X; C) [tD7*
*5, 7]. As pointed
out in [tD75, 7], see also [tD79, 5.3.11], the following diagram commutes
OE
A(G) ____//_C( G, Z)
oe|| r*||
fflffl|O fflffl|
R(G) ____//_C(aeG, C).
Lemma 9.2. Equivariant K-theory KG satisfies the 1-induction property.
*
* __
Proof.Since O is injective it suffices to note that the map r: aeG ! G factors*
* through A1. |__|
The Artin restriction theorem for equivariant K-theory follows from Theorem *
*5.7.
Proposition 9.3. For every G-spectrum X the restriction map
Q Q
KG(X) ! Eq[ AKA(X) ' H,L,gKH\gLg-1(X)]
is a |G|1-isomorphism. The first product in the equalizer is over representativ*
*es for conjugacy classes
of topologically cyclic subgroups of G, and the second product is over pairs of*
* these subgroups and
representatives g for the cosets HgL.
It suffices to pick a representative for each path component of the H-orbit *
*space of each sub-
manifold of G=L consisting of points with a fixed orbit type under the H-action.
19
Halvard Fausk
When G is connected, the maximal torus T is the only conjugacy class of subg*
*roups of G with
a dense subgroup generated by one element and with finite Weyl group by Example*
* 2.5. So when
G is connected the restriction map
KG(X) ! KT(X)WGT
is a |WGT |-isomorphism.
We give an explicit description of R(G) up to |G|1-isomorphism using the Art*
*in restriction
theorem. Let A* = hom(A, S1) denote the Pontrjagin dual of A. The elements of A*
** are the one
dimensional unitary representations of A. All irreducible complex representatio*
*ns of a compact
abelian Lie group are one dimensional. We verify that when A is a compact abeli*
*an Lie group the
canonical map
Z[A*] ! R(A)
is an isomorphism. A subgroup inclusion f: H ! L induces a restriction map f*: *
*L* ! H* of repre-
sentations. By the Artin restriction theorem we get that there is an (injective*
*) |WGT |-isomorphism
Q Q
R(G) ! Eq[ (A)Z[A*] ' (H),(L),gZ[(H \ gLg-1)*] ].
The Brauer induction theorem 6.5 and Lemma 9.2 applied to KG give that the i*
*dentity element
in R(G) can be induced up from the 1-hyper subgroups of G. This was first prove*
*d by G. Segal
[Seg68a, 3.11]. We get the following Brauer restriction theorem from Theorem 6.*
*6 and Lemma 9.2.
Proposition 9.4. For every G-spectrum X the restriction map
Q Q
KG(X) ! Eq[ H KH (X) ' H,L,gKH\gLg-1(X)]
is an isomorphism. The first product in the equalizer is over representatives f*
*or conjugacy classes of
1-hyper subgroups of G, and the second product is over pairs of these subgroups*
* and representatives
g for the cosets HgL.
There is another equivariant orthogonal K-theory KOG obtained by using real *
*instead of complex
G-bundles. It is a ring spectrum and the unit map of KG factors as
S0 ! KOG ! KG
where the last map is induced by tensoring the real G-bundles by C. The coeffic*
*ient ring of KOG
is RO(G) in degree 0. Since RO(G) injects into R(G) we get that KOG satisfies t*
*he 1-induction
property. Another example of a ring spectrum that satisfies 1-induction is Gree*
*nlees' equivariant
connective K-theory [Gre04].
Q
The restriction map R(G) ! R(C) is injective for all compact Lie groups G,*
* where the product
is over topological cyclic subgroups of G. When G is connected we even have tha*
*t R(G) ! R(T )WGT
is an isomorphism. Sometimes this can be used to prove stronger induction and d*
*etection type results
about KG than the results we get from our theory [Jac77, McC86]. See also Propo*
*sition 3.3 in [AS69 ].
An F-isomorphism theorem for equivariant K-theory has been proved by Bojanowska*
* [Boj83].
Remark 9.5. It is reasonable to ask if tom Dieck's equivariant complex cobordis*
*m spectrum satisfies
the induction property [tD70]. Morten Brun has shown that this is not the case.*
* In fact the unit
map A(G) ! MU0Gis injective when G is a finite group [Bru04].
References
AS69 M.F. Atiyah and G.B. Segal, Equivariant K-theory and completion, J. of di*
*fferential geometry 3,
1-18, 1969.
Ben95 D.J. Benson, Representation and cohomology I, Cambridge Studies in Advanc*
*ed Mathematics no. 30.
1995.
20
Generalized Artin and Brauer induction for compact Lie groups
Boj83 A. Bojanowska, The spectrum of equivariant K-theory, Math. Z. 183, no. 1,*
* 1-19, 1983.
Bre72 G.E. Bredon, Introduction to compact transformation groups, Academic Pres*
*s. 1972.
Bru04 M. Brun, Witt Vectors and Equivariant Ring Spectra, to appear in Proc. LM*
*S.
Con68 S. B. Conlon, Decompositions induced from the Burnside algebra, J. Algebr*
*a 10. 102-122, 1968.
tD70 T. tom Dieck, Bordism of G-manifolds and integrality theorems, Topology 9 *
*345-358, 1970.
tD75 T. tom Dieck, The Burnside ring of a compact Lie group I, Math. Ann.215. 2*
*35-250, 1975.
tD77 T. tom Dieck, A finiteness theorem for the Burnside ring of a compact Lie *
*group, Compositio Math.
35. 91-97, 1977.
tD79 T. tom Dieck, Transformation groups and representation theory, Lecture Not*
*es in Math. Vol 766.
Springer Verlag. 1979.
FO05 H. Fausk and B. Oliver, Continuity of ss-perfection for compact Lie groups*
*, Bull. London Math. Soc.
37. 135-140, 2005.
Fes79M. Feshbach, The transfer and compact Lie groups, Trans. Amer. Math. Soc. *
*251. 139-169, 1979.
Fes81M. Feshbach, Some general theorems on the cohomology of classifying spaces*
* of compact groups, Trans.
Amer. Math. Soc. 264. 49-58, 1981.
Gre04 J.P.C. Greenlees, Equivariant connective K-theory for compact Lie groups,*
* J. Pure Appl. Algebra 187
(2004), no. 1-3, 129-152.
Hat02 A. Hatcher, Algebraic topology, Cambridge University Press. 2002.
HKR00 M.J. Hopkins, N.J. Kuhn and D.C. Ravenel, Generalized group characters a*
*nd complex oriented
cohomology theories, J. Amer. Math. Soc. 13 553-594, 2000.
GM95 J.P.C. Greenlees and J.P. May, Generalized Tate cohomology, Memoirs of th*
*e Amer. Math. Soc.
Number 543. 1995.
Jac77S. Jackowski, Equivariant K-theory and cyclic subgroups, in C. Kosniowski,*
* Transformation groups,
London Math. Soc. no. 26. 76-91, 1977.
Lew96 L.G. Lewis, The category of Mackey functors for a compact Lie group, Grou*
*p representations: coho-
mology, group actions and topology (Seattle, WA, 1996), 301-354, Proc. Sympo*
*s. Pure Math., 63, Amer.
Math. Soc., Providence, RI, 1998.
LMS86 L.G. Lewis, J.P. May, and M. Steinberger (with contributions by J.E. McC*
*lure), Equivariant stable
homotopy theory, SLNM 1213. 1986.
May96 J.P. May, Equivariant homotopy and cohomology theories, CBMS. AMS. no.91.*
* 1996.
May01 J.P. May. Picard groups, Grothendick rings, and Burnside rings of categor*
*ies, Advances in Mathe-
matics 163, 1-16, 2001.
McC86 J.E. McClure, Restriction maps in equivariant K-theory, Topology Vol. 25*
*. No 4. 399-409, 1986.
MZ42 D. Montgomery and L. Zippin, Theorem on Lie groups, Bull. Amer. Math. Soc*
*. 48. 448-552, 1942.
Nis78G. Nishida, The transfer homomorphism in equivariant generalized cohomolog*
*y theories, J. Math.
Kyoto Univ. 18 , no. 3, 435-451, 1978.
Oli98B. Oliver, The representation ring of a compact Lie group revisited, Comme*
*nt Mat. Helv. 73, 1998.
Seg68aG.B. Segal, The representation ring of a compact Lie group, Publ. math. I*
*HES 34. 113-128, 1968.
Seg68bG.B. Segal, Equivariant K-theory, Publ. math. IHES 34. 129-151, 1968.
Sol67L. Solomon, The Burnside algebra of a finite group, J. Combin. Theory 2, 6*
*03-615, 1967.
Halvard Fausk fausk@math.uio.no
Department of Mathematics, University of Oslo, 1053 Blindern, 0316 Oslo, Norway
21