Henn, Hans-Werner

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AstérisqueHANS-WERNER HENNCohomological p-nilpotence criteria for compact Lie groupsAstérisque, tome 191 (1990), p. 211-220<http://www.numdam.org/item?id=AST_1990__191__211_0>© Société mathématique de France, 1990, tous droits réservés.L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/Publications/Asterisque/) implique l’accord avec les conditions générales d’uti-lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ouimpression systématique est constitutive d’une infraction pénale. Toute copieou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/COHOMOLOGICAL p-NILPOTENCE CRITERIA FOR COMPACT LIE GROUPS Hans-Werner Henn Introduction In [Ql] Quillen discussed cohomological criteria for p-nilpotence of fi­nite groups. He proved that for odd primes p a finite group G is j9-nilpotent if and only if the restriction map from the mod p cohomology H*(G;FP) to the mod p cohomology H*(GP;FP) of a p-Sylow subgroup Gp is an F-isomorphism. Recall that a map A B of graded F p algebras is called an F—isomorphism if and only if a G Kerny? implies an = 0 for some n and for each b 6 B some power is in the image of <p [Q2]. Furthermore Quil­len sketched a proof of the following result which he attributed to Atiyah: If p is any prime and J5P(G;FP) —* H*(GP;TFP) is an isomorphism for all sufficiently large i, then G is p—nilpotent. Quillen's main result in [Q2] can be interpreted as follows: For a com­pact Lie group G with classifying space BG the .F-isomorphism type of H*(BG;FP) is determined by the sets Rep(V, G) of G-conjugacy classes of homomorphisms from elementary abelian p-groups V to G [HLS]. In par­ticular, one can rephrase Quillen's p-nilpotence criterion in the following form: For an odd prime p a finite group G is j9-nilpotent if and only if inclusion induces a bijection Rep(V,Gp) —Rep(y, G) for all elementary abelian /^-groups V ([HLS; Prop. 4.2.3.]). If G is a compact Lie group with maximal torus T, normalizer NT, Weyl group W(G) = NT/T, then Gp will denote the preimage of Wp in NT. In this case Gp will be called a />-Sylow normalizer and is known to be a good substitute for a p-Sylow subgroup. S.M.F. Astérisque 191 (1990) 211 HENN In this paper we give for odd primes a characterization of those compact Lie groups G for which Rep(V, Gp) —• Rep(V, G) is a bijection for all V, or equivalently H*(BG]WP) —> H*(BGP;FP) is an F-isomorphism (Theorem 2.1.). The possibility of such a characterization was already mentioned in [HLS, Sect. 4.2.5.]. It seems appropriate to call such groups p-nilpotent compact Lie groups. We will also generalize Atiyah's criterion to the com­pact Lie group case (Theorem 2.5.). Our interest in such characterizations comes from the importance of BGP for the study of the (stable) homotopy type of BG. The paper is organized as follows. In section 1 we give the precise definition of a p-nilpotent compact Lie group and discuss some properties of such groups. We do not intend a systematic group theoretical study of this concept but will rather concentrate on properties which are relevant for our cohomological characterizations. These characterizations are stated and proved in section 2. The author would like to thank L. Evens and L. Schwartz for helpful discussions on this subject. This work was started while the author stayed at Northwestern University. He is grateful to the DFG and Northwestern University for supporting this stay and to the people at Northwestern for providing a pleasant and stimulating atmosphere. 1. p—nilpotent compact Lie groups 1.1 DEFINITION. A compact Lie group G is called p-nilpotent if and only if there is a finite normal subgroup N of order prime to p which together with Gp generates G. 1.2 REMARKS. (a) For finite groups this reduces to the classical definition of p-nilpotence. Then N consists of all elements of order prime to p and G/N is iso­morphic to Gp, i.e. G is a semidirect product iVxi Gp. In this case N is also called the normal p complement of Gp in G. (b) In the compact Lie group case G is in general not a semidirect product. For example, if G = (51, x,y \ [x.S1] = [j/,51] = x3 = y3 = 1, [a*, y] = £ with £ a primitive 3rd root of unity in 51) and p ^ 3, then 212 COHOMOLOGICAL P-NILPOTENCE OF LIE GROUPS Gp = S1 and the normal subgroup N = (x, y) shows that G is p-nilpotent. However, N H Gp ^ { 1 } and hence G ^ N>\ Gp. It is also obvious that G is not a semidirect product iVx Gp for some other N<G. Our definition of p-nilpotence above will be justified by the results below, which together with this example show that it would not be adequate to require the existence of a finite normal p-complement in the compact Lie group case. 1.3 PROPOSITION. Let G be a compact Lie group and p be any prime. Then the following statements are equivalent. (a) G is p-nilpotent. (b) Rep(<2, Gp) —1—> Rep(Q, G) is a bijection for all p-groups Q. (c) If Q is any finite p-subgroup of G, then NG(Q)/CG(Q), the quotient of the normalizer of Q in G by the centralizer of Q in G, is a finite p-group. (d) Each finite subgroup H of G is p-nilpotent. (e) G is a finite extension of a torus, i.e. there exists an exact sequence T • G » 7r with 7r finite, and G has a finite p-nilpotent subgroup H with H/H PI T = 7T and Tp = {t G T \ V = 1 } C H. (f) G is an extension of a torus by a finite p-nilpotent group TT and the conjugation action of the normal p-complement v of TTp in 7r is trivial on T. Proof, (a^ =» (b): Onto is equivalent to saying that any p-subgroup Q of G is conjugate to a subgroup of Gp, i.e. that the Q—set G/Gp has a nonempty <2-fixed point set (G/Gp)Q. This follows from x((G/Gp)«) = x(G/Gp) # 0 mod p where \ denotes Euler characteristic (cf. [HLS; Prop. 4.2.1.]). To show that i is 1 — 1 consider the projection Gp —Gp/Gp f] N = G/N. It suffices to show that 7r induces an injection on Rep(Q,?). So let ai, a2 be two homomorphisms with -KOL\ — g-KOL^g^1 for some g 6 Gp. By factoring out the kernel we may assume that irai is mono. Identify Q with its image in Gp/GpnN. Then a± and g^g"1 are sections of 7r~1(Q) Q. Now Kern7r = Gp D N is a subgroup of T of order prime to p and hence 213 HENN i71(Q,Gp n N) = 0, i.e. ai and got^g 1 are even conjugate by an element in Gp H N and we are done. (b) =» (c): For any group G the automorphism group Aut(Q) acts on Rep(Q, G). If Q is a subgroup of G, then NG(Q)/CG (Q) identifies naturally with the isotropy subgroup of the inclusion Q c—• G, considered as an element in the Aut(Q)-set Rep(Q,G). Now (b) implies that we can assume that Q is a subgroup of Gp and that it suffices to show that NGP{Q) / C GP{Q) is ap-group. So suppose that x G NGP {Q) has order prime to p in NGP(Q)/CGP(Q)- AS in [HLS, sect. 4.3.] we may assume that x itself has order prime to p, i.e. x G T. Then one sees as in [HLS, Lemma 4.3.3.] that x acts trivially on the quotient of Q by its Frattini-subgroup (f> (Q) and hence trivially on Q (cf. [H, Satz III 3.18.]). Therefore x is in CGP(Q) and we are done. (c) (d): If Q is a subgroup of H, then iV/f (Q)/G# (Q) is a subgroup of NG(Q)/CG(Q) and hence the Frobenius criterion [H, Satz IV, 5.8.] implies that H is p-nilpotent. For the remaining implications we need a Lemma. For a natural number £ let Te denote {t G T \ te = 1}. 1.4 LEMMA. Let G be an extension of a torus T by a finite group TT of order \7r\. Then there is a finite subgroup F of G with F/F D T = 7r and F n T = TM. Proof. Interpret the (class of the) extension T ^ G » n as an element [e] G H2(TT;T) and use that \TT\ - [e] = 0 together with the long exact coho-mology sequence arising from the short exact sequence Tĵ i T ^>> T of 7r-modules. • We continue with the proof of Proposition 1.3. (d) =» (e): Assume that G is not a finite torus extension. Then G(x), the connected component of 1, is not abelian and hence contains a compact connected nonabelian Lie group of rank 1, i.e. either 50(3) or SU{2). Now 50(3) contains A4, the alternating group on four letters, as symmetry group 214 COHOMOLOGICAL P-NILPOTENCE OF HE GROUPS of a regular tetrahedron. As neither A4 nor its twofold cover in SU{2) are 2-nilpotent, we may assume that p is odd. Next consider G := NTnG(i). This is a finite torus extension, so there is a finite subgroup F as in Lemma 1.4. Let H be the finite subgroup of G, generated by F and Tp (finite because Tp is normal). If G(i) ^ T, then the Weyl group W(G(i)) is nontrivial. Pick a reflection in W(G(i)) and represent it by an element r £ H. Then r defines a nontrivial element of order 2 in N~(TP)/C~(TP) and hence H is not p-nilpotent. We conclude that G(i) is a torus and G is a finite torus extension. Now let F C G be as in 1.4. Then H = (F, Tp) is the finite group with the desired properties. (e) =>> (f): If N is the normal p complement of Hp in H, then N/N H T is the normal p complement of 7rp in 7r. Therefore it suffices to show that N commutes with T. Now N and Tp are both normal in H and have trivial intersection, hence they commute. Finally, a smooth automorphism of T which fixes Tp is clearly trivial, if p is odd, or has order at most 2, if p = 2. Hence N commutes with T and we are done. (f) => fa): Let G' be the preimage in G of the normal p complement v. Then Lemma 1.4 gives a subgroup Ff of Gf with.F'/Ff C\T = v and F' OT = T\v\, where \v\ is the order of z/. Clearly, is a finite group of order prime to p which together with Gp generates G. However, F1 need not be normal. Therefore consider the subgroup N = (F'^T^*) C G. This is still a finite group of order prime to p. We claim that iV is normal. For this it suffices to show that gFrg~x C N for all g G G. So let x be in F'. Then gxg"1 = yt for some y € i7", t G T, since z/ is normal in 7r. It suffices to show that *M = 1 . This follows because the order of elements in F' clearly divides \v\2 and because y commutes with t by assumption. This finishes the proof of 1.3. • 2. Cohomological p— nilpotence criteria Before we state our main result we recall that a subgroup V of Gp is said to be weakly closed in Gp with respect to G if gVg"1 C Gp, g € G, implies ffT^ff""1 = V. 215 HENN 2.1 THEOREM. Let G be a compact Lie group and p be an odd prime. Then the following statements are equivalent. (a) G is p-nilpotent. (b) Rep(V, Gp) —» Rep(V, G) is bijective for all elementary abelian p-groups V. (c) Let V be any normal elementary abelian p-subgroup of Gp which contains Tp. Then V is weakly closed in Gp with respect to G and NG(V)/CG(V) is a finite p-group. 2.2 REMARKS. (a) We recall that condition 2.1.(b) is equivalent to the map H*(BG;FP) —• H*(BGp;Fp) being an F isomorphism. In fact, a transfer argument shows that this map is mono for all compact Lie groups G. If G is also p-nilpotent then the Leray-Serre spectral sequence of the fibration B(N PI Gp) -+ BGP -> B{Gp/Gp n N) = B(G/N) with mod p acyclic fibre shows that H*(BG;FP) —> H*(BGpjFp) is also onto and hence a genuine isomorphism. (b) In the finite case condition 2.1.(c) above gives just Quillen's group theo­retical version of his p-nilpotence criterion ([Ql, Thm. 1.5.]). The proof of implication (c) (a) below is essentially a careful modification of the proof of Theorem 1.5. in [Ql]. (c) For p = 2 there are examples of compact Lie groups G which satisfy conditions 2.1.(b) and 2.1.(c) but which are not 2-nilpotent. G = SU(2) is an example of a connected group and G = Qs X Z /3 the semi direct product of the quaternion group with ZZ/3 (cf. [Ql]), is an example of a finite group. A cohomological criterion for p-nilpotence that works for all primes will be given below in Theorem 2.5. Proof of Theorem 2.1. (a) =» (b): This follows from Proposition 1.3. (b) (c): Clearly, (b) implies that a normal elementary abelian p-subgroup V of Gp is weakly closed with respect to G. The proof of Propo­sition 1.3. ((b) (c)) shows that NG(V)/CG(V) is a p-group. 216 COHOMOLOGICAL P-NILPOTENCE OF HE GROUPS (c) =» (a): If G is not a finite torus extension, then we see as in the proof of Proposition 1.3. ((d) (e)) that NG(TP)/CG(TP) contains a nontrivial element of order 2 in contradiction to our assumptions. Therefore G is a finite torus extension. Denote G/T by 7r and let F be a finite subgroup of G with T fl F = T\n\ and F/F fl T = 7r as in Lemma 1.4. By criterion (e) of Proposition 1.3. it suffices to show that the finite group H = (F,TP) is p-nilpotent. We pick a p-Sylow subgroup Hp of H which is contained in Gp. 2.3 LEMMA. Let V be any abelian subgroup of H (resp. Hp) which contains Tp. Then V is normal in H (resp. Hp) if and only ifV is normal in G (resp. Gp), provided p is odd. Proof. Suppose V is abelian and contains Tp. Then V commutes with Tp and hence with T (p is odd!). Therefore, if H normalizes V, then (i?, T) = G normalizes V. Similarly with Hp and Gp. The converse is trivial. • We return to the proof of 2.1. ( (c) (a) ) Lemma 2.3 implies that any normal elementary abelian p-subgroup V of Hp containing Tp is weakly closed in Hp with respect to H. Furthermore, NH(V)/CH(V) is a subgroup of NG(V)/CG(V), in particular a p-group. Therefore, the p-nilpotence of H is a consequence of the following slight generalization of Quillen's Theorem 1.5. in [Ql]. 2.4 PROPOSITION. Let p be an odd prime and G be a finite group with p-Sylow subgroup Gp. Let U be a normal elementary abelian p-subgroup of G and assume that each normal elementary abelian p-subgroup V of Gp con­taining U is weakly closed in Gp with respect to G and that NG(V)/CG(V) is a p-group for such V. Then G is p-nilpotent. Proof of 2.4. The proof is almost the same as in [Ql]. For the convenience of the reader we repeat the main steps. The hypothesis of 2.4. are inherited by all subgroups of G which contain Gp. Therefore we can do induction on the order of such subgroups. 217 HENN Let V be a subgroup of Gp which contains U and is maximal with respect to being elementary abelian and normal in Gp. Then V is a maximal elementary abelian subgroup of G (cf. [Ql, Prop. 4.1.]) and hence CG{V) is p-nilpotent by [H, Satz IV, 5.5.]. Now there are two cases: Case 1: V is normal in G. Then G is p-nilpotent because CG(V) is p-nilpotent and G/CG{V) = NG(V)/CG(V) is a p-group. Case 2: V is not normal in G. Then let W be a maximal G-normal subgroup of V which contains U. Define subgroups V\ of V and N of G by Vx/W = V/W H Z(GP/W) (Z denotes the center) N = NG(V1). Then everything works precisely as in [Ql]. - N contains Gp and is properly contained in G, hence N is p-nilpotent by induction. - Vi/W is a central subgroup of Gp/W which is weakly closed with respect to G/W. Therefore, Grun's Theorem implies HX(G/W) -̂ U HX{N/W) and the cohomology 5-term exact sequences of the group extensions W G » G/W, W ^ N » N/W yield HX{G) HX{N) . - Finally, Tate's if1-criterion [T] implies that G is p-nilpotent. • • The following result generalizes Atiyah's p-nilpotence criterion and is valid for all primes. 2.5 THEOREM. Let G be a compact Lie group and suppose inclusion in­duces an isomorphism Hl(BG; Fp) Hl(BGp; Fp) for all sufficiently large i. Then G is p-nilpotent. Proof. By a transfer argument (cf. [CI] for the existence of a stable transfer map) there is a p-local stable splitting BGP^BG V X for some p-local connected X with bounded above and finite type mod p homology. Now Gp is a finite torus extension. Let F be a finite subgroup of Gp as in Lemma 1.4. If Tpoo denotes the subgroup of T consisting of all torsion elements 218 COHOMOLOGICAL P-NILPOTENCE OF HE GROUPS of order a power of p, then the inclusion (Tpoo, F) «—• Gp induces a mod p homology equivalence and therefore there is for each n a finite p-subgroup Fn of (Tpoo^F) such that inclusion induces an epimorphism Hi(BFn;Fp) —• Hi(BGp]lFp) for all i < n. In particular, there exists n such that there is a stable map BFn —• X (after localizing at p) which is onto in mod p homology. Now the solution of the Segal conjecture [Ca] forces X to be trivial because there are no nontrivial stable maps from BFn to any positive dimensional sphere. We conclude that Hl(BG;Fp) —> Hl(BGp;Fp) is an isomorphism for all i. For 2 — 1 we get (2.6) Jf^JSGjFp) S Hom(#iOBGO;Fp) ^ Hom(TTiCBG);Fp) ^ Hom(7r0(G);Fp) and therefore we have a bijection (2.7) Hom(7r0(G);Fp) Hom(7r0(Gp);Fp). Because of Theorem 2.1 (cf. remark 2.2) we may assume p = 2. The determinant of the adjoint representation of a 2-Sylow normalizer G2 on the Lie algebra LT defines a homomorphism ^(G^) F2. If T is properly contained in G1, the connected component of 1 G G, then the reflections in the Weyl group W{G^) show that (p restricts nontrivially to 7To(Cr2 nG(i)) and can therefore not come from 7To(G) . It follows that T = G(i) and G is a finite torus extension. Now (2.6), (2.7) and Tate's iJ1-criterion imply that 7r0(G) = G/T is 2-nilpotent. By Proposition 1.3.(f) it suffices therefore to show that odd order elements of 7TQ(G) act trivially on T. Our hypothesis implies certainly that H*(BG;WP) —• H*(BGP;FP) is an i^-isomorphism, hence Rep(V, Gp) —+ Rep(V, G) is bijective for all elementary abelian p-groups V and therefore NG(TP)/CG(TP) is ap-group by the proof of Proposition 1.3.((b) (c)). For p = 2 it follows that odd order elements of TT0(G) act trivially on T2 and hence on T (cf. proof of Proposition 1.3. ((e) (f)) ). This finishes the proof of 2.5. • 219 HENN REFERENCES [Ca] G. Carlsson, Equivariant stable homotopy and Segal's Burnside ring conjecture, Ann. of Math. 120 (1984), 189-224 [Cl] M. Clapp, Duality and transfer for parametrized spectra, Archiv der Mathematik 37 (1981), 462-472 [HLS] H.-W. Henn, J. Lannes and L. Schwartz, Analytic functors, unstable algebras and cohomology of classifying spaces, Contemporary Mathematics 96 (1989), 197-220 [H] B. Huppert, Endliche Gruppen I, Grundlehren der mathematischen Wissenschaften 134, Springer 1967 [Q1] D. Quillen, A cohomological criterion for p-nilpotence, J. P. A. A. 1 (1971), 361-372 [Q2] D. Quillen, The spectrum of an equivariant cohomology ring I, Ann. of Math. 94 (1971), 549-572 [T] J. Tate, Nilpotent quotient groups, Topology 3 (1964), 109-111 Hans—Werner Henn Mathematisches Institut der Universität Im Neuenheimer Feld 288 D-6900 Heidelberg Federal Republic of Germany 220 

Cohomological $p$-nilpotence criteria for compact Lie groups

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Cohomological $p$ -nilpotence criteria for compact Lie groups

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Cohomological ppp-nilpotence criteria for compact Lie groups

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Cohomological $p$ -nilpotence criteria for compact Lie groups