THE RATIONAL COHOMOLOGY OF A p-local COMPACT GROUP

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1 THE RATIONAL COHOMOLOGY OF A -LOCAL COMPACT GROUP C. BROTO, R. LEVI, AND B. OLIVER Let be a rime number. In [BLO3], we develoed the theory of -local comact grous. The theory is modelled on the -local homotoy theory of classifying saces of comact Lie grous and -comact grous, and generalises the earlier concet of -local finite grous [BLO2]. It rovides a coherent context in which classifying saces of comact Lie grous and -comact grous [DW] can be studied, and also gives rise to many exotic examles. In this aer, we study the rational -adic cohomology H Q ( ) def = H (, Z ) Q of a -local comact grou. Our main result here is that, as one would exect, the -adic rational cohomology of -local comact grous behaves the same way as that of a comact Lie grou. Theorem A. Let G = (S, F, L) be a -local comact grou. Let S 0 S be its maximal torus, and let W (G) def = Aut F (S 0 ) be its Weyl grou. Then H Q (BG) = H Q (BS 0 ) W (G). Preublicació Núm 7, febrer Deartament de Matemàtiques. htt:// Of course, the Weyl grou of a -local comact grou need not be a seudo-reflection grou, and hence the rational cohomology of the classifying sace is not in general a olynomial algebra. Like comact Lie grous and -comact grous, -local comact grous admit unstable Adams oerations, which are defined in [JLL], using the internal structure of the -local grou in question, rather than its rational cohomology. One alication of Theorem A is Proosition 3.2, which states that under a mild condition, the obvious cohomological definition of an unstable Adams oeration characterises the same family of mas as the one referred to in [JLL] as geometric unstable Adams oerations. Another easy alication of Theorem A is the observation that if G is a -local comact grou with maximal torus S 0, then the inclusion in G of the -local subgrou given by the normaliser N G (S 0 ) induces a rational -adic cohomology isomorhism. In Section 1, we recall the basic concets in the theory of -local comact grous which will be needed to rove Theorem A. Section 2 is dedicated to the roof of the theorem. Finally in Section 3 we discuss the alications described above Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20. Key words and hrases. Classifying sace, -comletion, -local comact grous, fusion. C. Broto is artially suorted by FEDER-MICINN grant MTM R. Levi is artially suorted by EPSRC grant EP/I019073/1. B. Oliver is artially suorted by UMR 7539 of the CNRS, and by roject ANR BLAN , HGRT. 1

2 2 C. BROTO, R. LEVI, AND B. OLIVER 1. Some basic concets We recall the definition and some basic roerties of -local comact grous. The reader is referred to [BLO3] for a comrehensive account of these objects. We begin by defining discrete -toral grous. By Z/ we mean the union of all Z/ r with resect to the natural inclusions. Definition 1.1. A discrete -torus is a grou isomorhic to (Z/ ) r for some ositive integer r. A discrete -toral grou is a grou S which contains a normal discrete -torus S 0, with -ower index. The normal subgrou S 0 will be referred to as the maximal torus or the identity comonent of S, and the quotient grou Γ = S/S 0 will be called the grou of comonents of S. The identity comonent S 0 of a discrete -toral grou S can be characterised as the subset of all infinitely -divisible elements in S, and also as the unique minimal subgrou of finite index in S. Thus, S 0 is a characteristic subgrou. The rank of S is the number r = rk(s) such that S 0 = (Z/ ) r. Recall that for P, Q S, the transorter set T S (P, Q) is the set of all elements g S such that gp g 1 Q. We denote by Hom S (P, Q) the set of all homomorhisms c g : P Q, which are restrictions of an inner automorhism of S, and by Inj(P, Q) denote the set of all the injective homomorhisms P Q. We are now ready to recall the definition of fusion systems over discrete -toral grous. Definition 1.2. A fusion system F over a discrete -toral grou S is a category whose objects are the subgrous of S, and whose morhism sets Hom F (P, Q) satisfy the following conditions: (a) Hom S (P, Q) Hom F (P, Q) Inj(P, Q) for all P, Q S. (b) Every morhism in F factors as an isomorhism in F followed by an inclusion. Two subgrous P, P S are called F-conjugate if P and P are isomorhic as objects in F. A subgrou P S is said to be F-centric if for every subgrou P S which is F-conjugate to P, C S (P ) = Z(P ). All fusion systems considered in this aer are required to be saturated [BLO3, Definition 2.2]. Although the results we resent here are based on roerties of saturated fusion systems roved in [BLO3], we do not exlicitly use the saturation axioms, and thus we will not reeat them here. Next, we briefly recall what are centric linking systems and -local comact grous. The full definition can be found in [BLO3, Definitions 4.1, 4.2] Definition 1.3. Let F be a fusion system over a discrete -toral grou S. A centric linking system associated to F is a category L whose objects are the F-centric subgrous of S, together with a functor π : L F c, and distinguished monomorhisms P δ P Aut L (P ) for each F-centric subgrou P S, which satisfy the following conditions. (A) π is the identity on objects and surjective on morhisms. More recisely, for each air of objects P, Q L, the centre Z(P ) acts freely on Mor L (P, Q) by comosition (uon identifying Z(P ) with δ P (Z(P )) Aut L (P )), and π induces a bijection Mor L (P, Q)/Z(P ) = Hom F (P, Q).

3 THE RATIONAL COHOMOLOGY OF A -LOCAL COMPACT GROUP 3 (B) For each F-centric subgrou P S and each g P, π sends δ P (g) Aut L (P ) to c g Aut F (P ). (C) For each f Mor L (P, Q) and each g P, the following square commutes in L: δ P (g) P P f f Q Q. δ Q (π(f)(g)) A -local comact grou is a trile G = (S, F, L), where S is a discrete -toral grou, F is a saturated fusion system over S, and L is a centric linking system associated to F. The classifying sace of G is the -comleted nerve L, which we will generally denote by BG. In [BLO3], the authors show that comact Lie grous and -comact grous give rise to articular examles of -local comact grous. Another large family of examles arises from linear torsion grous. In each case, the resective classifying sace coincides u to homotoy (after -comletion in the case of genuine grous) with the classifying sace of the -local comact grou it gives rise to. Definition 1.4. Let G = (S, F, L) be a -local comact grou. Then the Weyl grou W (G) of G is defined to be the automorhism grou in F of the maximal torus S 0 S. Notice that HQ (X) def = H (X, Z ) Q is not in general isomorhic to H (X, Q ). For instance if X = BZ/, then HQ (X) is a olynomial ring over the -adic rationals on a generator in degree 2, while H (X, Q ) is trivial. The use of HQ as the aroriate cohomology theory for our urose goes back to Dwyer and Wilkerson [DW], in their first aer on -comact grous. 2. The rational cohomology Two rearatory lemmas are needed before we rove our main claim. Lemma 2.1. Let P be a discrete -toral grou with maximal torus P 0 P. Then H Q (BP ) = H Q (BP 0 ) P/P 0. Proof. This is of course a articular case of a much more general statement. U to homotoy, BP 0 is a covering sace of BP with grou P/P 0, and so one has the usual transfer ma Tr: H (BP 0, Z ) H (BP, Z ), where Tr Res is multilication by P/P 0. Hence after tensoring with Q this comosite is an isomorhism. On the other hand, the comosition the other way Res Tr is norm ma for the action of P/P 0 on HQ (BP 0 ), and hence the image of restriction is the subgrou of invariants HQ (BP 0 ) P/P 0. To rove the theorem, we will use the subgrou decomosition for -local comact grous [BLO3, Proosition 4.6]. Hence the following lemma is an essential ingredient. In order to state it, we need to recall some notation and terminology. For a fusion system F over a discrete -toral grou S, we denote by O(F) the orbit category associated to F, i.e., the category with the same objects and with

4 4 C. BROTO, R. LEVI, AND B. OLIVER morhisms Mor O(F) (P, Q) = Re F (P, Q) def = Hom F (P, Q)/Inn(Q). For P, Q S, let N S (P, Q) denote the transorter set consisting of all elements of S which conjugate P into Q. If F is a full subcategory of F, we denote by O(F ) the full subcategory of O(F) whose objects are those of F. If Γ is a finite grou, we denote by O (Γ) the category whose objects are the -subgrous of Γ and whose morhisms are Mor(P, Q) = C Γ (P )\N Γ (P, Q)/Inn(Q). Lemma 2.2. Let F be any saturated fusion system over a discrete -toral grou S. Define F : o Q-mod on objects by setting F (P ) = HQ (BP ). On morhisms, F sends the class of ϕ Hom F (P, P ) to the homomorhism induced by Bϕ. Then F is acyclic, namely lim i (F ) = 0 for all i > 0. Proof. Set Q = C S (S 0 ) S, and Γ = Out F (Q). Then Q is F-centric, and is weakly closed in F since S 0 is. Let F Q denote the full subcategory of F whose objects are those P S which contain Q, and let Θ: O(F Q ) O (Γ) be the functor which sends an object P to Out P (Q) Γ, and a morhism ϕ Re F (P, P ) to the class of ϕ Q N Γ (Θ(P ), Θ(P )) (see [BLO3, Lemma 5.7]). For each -subgrou Π Γ, regarded as a grou of automorhisms of S 0, define Φ (Π) = H Q (BS 0 ) Π. This defines a graded functor Φ : O (Γ) o Q-mod. Furthermore, for each P S which contains Q, F (P ) = HQ (BQ) P/Q = Φ (Θ(P )). Thus Φ Θ = F O(F Q ). For each P S, Out Q (P ) acts trivially on F (P ) since Q centralises P 0, and F (P ) is a subring of H Q (BP 0 ). So by [BLO3, Lemma 5.7], lim (F ) = lim (Φ ). O (Γ) The functor Φ is a Mackey functor on O (Γ), and hence is acyclic (see [JM, Proosition 5.14] or [JMO, Proosition 5.2]). We are now ready to rove our main theorem. Theorem 2.3. Let G = (S, F, L) be a -local comact grou. Then H Q (BG) = H Q (BS 0 ) W (G). Proof. Let π : L be the rojection, and let B : To denote the left homotoy Kan extension of the constant functor on L along π. Then there is a homotoy equivalence hocolim B L, and for each object P, B(P ) BP [BLO3, Proosition 4.6]. Consider the Bousfield-Kan sectral sequence [BK] for cohomology of the homotoy colimit, with

5 THE RATIONAL COHOMOLOGY OF A -LOCAL COMPACT GROUP 5 coefficients in the -adic integers Z. Since Q is flat as a Z-module, one can tensor the sectral sequence with Q to get a sectral sequence for -adic rational cohomology E,q 2 = lim H q Q ( B( )) = H +q Q ( L ). By Lemma 2.2, the higher limits all vanish and we obtain the formula HQ ( L ) = lim HQ ( B( )). (1) For each F-centric subgrou P S, let ι P : P S denote the inclusion. The inverse limit in (1) consists of all elements x HQ (BS) such that ϕ ι Q (x) = ι P (x) for all F-centric subgrous P, Q S, and all morhisms ϕ Hom F (P, Q). Let ϕ: P Q be any morhism in F, where P and Q are F-centric. Then by [BLO3, Lemma 2.4] the restriction ϕ P0 coincides with the restriction to P 0 of some automorhism σ W (G). Let x HQ (BS 0 ) W (G) HQ (BS) be any element. Then ι P (x) H Q (BP ) HQ (P 0 ), and ι Q (x) H Q (BQ) HQ (BQ 0 ), and Hence ϕ (ι Q(x)) = σ (ι Q(x)) = ι P σ (x) = ι P (x). HQ (BS 0 ) W (G) lim H Q ( B( )). Conversely, let y HQ (BS) HQ (BS 0 ) be an element which is stable under each morhism in F between centric subgrous, and let σ W (G). By Alerin s fusion theorem, σ can be decomosed into a sequence σ = σ 1 σ 2 σ n, where each σ i W (G) can be extended to an automorhism of some F-centric subgrou P i S. But since y is stable under each of the σi, it is also stable under σ. This shows that and thus comletes the roof of our claim. lim H Q ( B( )) = HQ (BS 0 ) W (G) 3. Alications For a comact Lie grou G, one defines an unstable Adams oeration of degree ζ to be a selfma of the classifying sace inducing multilication by ζ i on rational cohomology in dimension 2i, where ζ is an integer. An analogous definition is made for -comact grous, excet ζ is required to be a -adic unit, and rational cohomology is relaced by -adic rational cohomology. Unstable Adams oerations are a very imortant concet in the homotoy theory of classifying saces of comact Lie grous and -comact grous. In [JLL], it is shown that -local comact grous also admit unstable Adams oerations. Let G = (S, F, L) be a -local comact grou and let ζ be a -adic unit. A normal Adams automorhism of degree ζ on S is an automorhism φ Aut(S) which restricts to the ζ-ower ma on S 0, and induces the identity on the grou of comonents S/S 0. A geometric unstable Adams oeration of degree ζ on G is a selfma Ψ of BG, such that there exist a normal Adams automorhism φ of degree ζ on S, with the roerty that ι Bφ Ψ ι. Here ι: BS BG is the canonical inclusion. (See [JLL, Definitions 2.3, 3.4]) Theorem A allows us to define geometric unstable Adams

6 6 C. BROTO, R. LEVI, AND B. OLIVER oerations of -local comact grous, along the lines of the classical cohomological definition. The following lemma is an analogue of a theorem of Notbohm [N, Proosition 4.1]. Lemma 3.1. Let G = (S, F, L) be a -local comact grou, and let T be a discrete -torus. Then there is an isomorhism Hom(T, S 0 )/W (G) = [BT, BG], where W (G) acts by left translation. Also, two mas f, h: BT BG are homotoic if and only if they induce the same homomorhism on H Q ( ). Proof. By [BLO3, Theorem 6.3 (a)] there is an isomorhism of sets Re(T, L) = [BT, BG], where Re(T, L) def = Hom(T, S)/, with α β if and only if there is some ϕ Hom F (α(t ), β(t )) such that ϕ α = β. Since T is a discrete -torus, the image of every homomorhism from it to S is contained in S 0, and by [BLO3, Lemma 2.4 (b)], every homomorhism in F between subgrous of S 0 is the restriction of some element in W (G). Thus as claimed. Re(T, L) = Hom(T, S 0 )/W (G), It remains to rove the last statement. Two mas f, h: BT BG that are homotoic clearly induce the same ma on cohomology. Conversely, assume that f, h: BT BG are two mas such that f = g : H Q (BG) H Q (BT ). Let α, β : T S 0 be homomorhisms such that f = ι Bα and g = ι Bβ, where ι: BS 0 BG is the inclusion of the maximal torus. We will show that there is w W (G) such that w α = β, following the argument used by Adams and Mahmud to rove [AM, Theorem 1.7]: an argument based on the uniqueness of factorisation in the olynomial ring H Q (BS 0 ). For simlicity, write V = H 2 Q (BS 0 ) and V = H 2 Q (BT ). For each w W (G), define V (w) = { x V Bβ (x) = B(w α) (x) } = Ker ( (Bβ B(w α) ) V ). For each x V, set x = w W (G) Bw (1 + x) S(V ) = H Q (BS 0 ) where S(V ) denotes the symmetric algebra on the Q -vector sace V. Since x is W (G)- invariant, Theorem 2.3 imlies that x Im(ι ), and hence that Bα ( x) = Bβ ( x). In other words, (1 + Bα Bw x) = (1 + Bβ Bw x) S(V ). w W (G) w W (G) Since S(V ) is a unique factorization domain, there is w W (G) such that (1+Bβ x) = λ(1 + Bα Bw x), for some λ Q. Then λ = 1 and hence Bβ x = Bα Bw x. In articular, x V (w). This roves that V = w W (G) V (w). Since Q is infinite, V finite dimensional, and W (G) finite, there is w W (G) such that V = V (w) (cf. [AM, Lemma 3.1]). Hence Bβ = B(w α). Since Hom(T, S 0 ) injects into Hom ( HQ (BS 0 ), HQ (T ) ), it now follows that w α = β Hom(T, S 0 ), and hence that f g as mas BT BG.

7 THE RATIONAL COHOMOLOGY OF A -LOCAL COMPACT GROUP 7 Proosition 3.2. Let G = (S, F, L) be a -local comact grou, and let ζ be a -adic unit. Then any geometric unstable Adams oeration Ψ of degree ζ induces multilication by ζ i on HQ 2i (BG). If S 0 is self centralising in S, then any self equivalence Ψ of BG which induces multilication by ζ i on HQ 2i (BG) for each i is a geometric unstable Adams oeration on F. Proof. Let ι: BS BG be the canonical inclusion (induced by the distinguished monomorhism δ S : S Aut L (S)), and set ι 0 = ι S0. If ψ is a geometric unstable Adams oeration on G of degree ζ, then by definition, there exists a normal Adams automorhism φ of S such that Ψ ι ι Bφ, and hence Ψ ι 0 ι 0 B(φ S0 ). For each i 0, φ S0 induces multilication by ζ i on HQ 2i (BS 0 ), and hence Ψ does the same on HQ 2i (BG). Assume now that S 0 is self centralising in S. Let Ψ: BG BG be a self equivalence such that Ψ is multilication by ζ i on HQ 2i (BG). By [BLO3, Theorem 6.3(a)] and Lemma 3.1, the natural mas End(S)/ Aut F (S) = [BS, BG] and End(S 0 )/W (G) = [BS 0, BG] (2) are bijections. Hence there is ϕ End(S) such that ι Bϕ Ψ ι, and ϕ Aut(S) since Ψ is a homotoy equivalence. Let ϕ 0 Aut(S 0 ) be the restriction of ϕ to S 0, let ζ denote the ζ-ower ma on S 0, and set ρ = ζ ϕ 1 0 Aut(S 0 ). Then Bϕ 0 ι 0 = B ζ ι 0 : H Q (BG) H Q (BS 0 ), and by Lemma 3.1, there is w W (G) such that w ϕ 0 = ζ Aut(S 0 ). Fix a morhism ι Mor L (S 0, S) such that π( ι) is the inclusion, and regard this as the inclusion of S 0 in S in the category L. By [BLO3, Lemma 4.3(a)], for each g S, there is a unique restriction δ S0 (g) Aut L (S 0 ) of δ S (g) Aut L (S); i.e., a unique morhism such that ι δ S0 (g) = δ S (g) ι. Identify S and S 0 with their images in Aut L (S 0 ). Let α Aut L (S 0 ) be a lift of w, i.e., π(α) = w. By Axiom (C), for each t S 0, α δ S0 (t) = δ S0 (w(t)) α. Hence, c α S0 = w, and so χ def = c α S ϕ: S Aut L (S 0 ) restricts to w ϕ 0 = ζ on S 0. Now, for each g S, χ c g = c χ(g) χ as automorhisms of S 0, and since χ S0 = ζ is central in Aut(S 0 ), c g S0 = c χ(g) S0. Since S 0 is self centralising in S, it follows that for each g S, g χ(g) (mod S 0 ). In articular, χ(s) = S, and χ induces the identity on S/S 0. Thus χ is a normal Adams automorhism of S of degree ζ. Also, ι Bχ ι B(c α S ) Bϕ ι Bϕ Ψ ι, and thus Ψ is a geometric unstable Adams oeration on G as claimed. If G = (S, F, L) is a -local comact grou, and P S is a subgrou satisfying a certain mild condition (fully normalised), then one can define the normaliser fusion system, N F (P ), which is shown in [BLO6, Theorem 2.3] to be a saturated fusion system. The normaliser linking system N L (P ) can be defined in exactly the same way as in [BLO2, Definition 6.1], and the roof of [BLO2, Lemma 6.2] alies verbatim to show that N L (P ) is a centric linking system associated to N F (P ). Thus in this case N G (P ) = (N S (P ), N F (P ), N L (P )) is a -local comact subgrou of G. In articular, the maximal torus S 0, is fully normalised, since it is unique in its F-conjugacy class, and we may consider the inclusion N G (S 0 ) G. (3)

8 8 C. BROTO, R. LEVI, AND B. OLIVER Then, S 0 is the maximal torus in both G and N G (S 0 ), and from the definition of morhisms in the normaliser fusion system [BLO6, Definition 2.1], W (G) = Aut F (S 0 ) = Aut NF (S 0 )(S 0 ) = W (N G (S 0 )). Thus one obtains as an immediate corollary of Theorem A, that the inclusion (3) induces an isomorhism in -adic rational cohomology. This is analogous to the corresonding statements for comact Lie grous and -comact grous. References [AM] J.F. Adams & Z. Mahmud, Mas between classifying saces, Inv. Math. 35 (1976), 1 41 [BK] P. Bousfield & D. Kan, Homotoy limits, comletions, and localizations, Lecture notes in math. 304, Sringer-Verlag (1972) [BLO2] C. Broto, R. Levi, & B. Oliver, The homotoy theory of fusion systems, Journal Amer. Math. Soc. 16 (2003), [BLO3] C. Broto, R. Levi, & B. Oliver, Discrete models for the -local homotoy theory of comact Lie grous and -comact grous, Geom. & Tool. 11 (2007), [BLO6] C. Broto, R. Levi, & B. Oliver, In Prearation. [DW] W. G. Dwyer, & C. W. Wilkerson, Homotoy fixed-oint methods for Lie grous and finite loo saces. Ann. of Math. (2) 139 (1994), no. 2, [JLL] F. Junod, R. Levi and A. Libman, Unstable Adams oerations of -local comact grous, [JM] Alg. and Geom. Tool. 12 (2012) 4974 Stefan Jackowski, J. McClure, Homotoy decomosition of classifying saces via elementary abelian subgrous. Toology 31 (1992), no. 1, [JMO] S. Jackowski, J. McClure, & B. Oliver, Homotoy classification of self-mas of BG via G- actions, Annals of Math. 135 (1992), [N] D. Notbohm, Abbildungen zwischen klassifizierenden Räume, Dissertation, Göttingen, 1988 Deartament de Matemàtiques, Universitat Autònoma de Barcelona, E Bellaterra, Sain address: broto@mat.uab.es Deartment of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K. address: ran@maths.abdn.ac.uk LAGA, Institut Galilée, Av. J-B Clément, Villetaneuse, France address: bob@math.univ-aris13.fr