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]. 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