A FORMULA FOR p-completion BY WAY OF THE SEGAL CONJECTURE

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1 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE SUNE PRECHT REEH, TOMER M. SCHLANK, AND NATHANIEL STAPLETON Abstract. The Segal conjecture describes stable mas between classifying saces in terms of (virtual) bisets for the finite grous in question. Along these lines, we give an algebraic formula for the -comletion functor alied to stable mas between classifying saces urely in terms of fusion data and Burnside modules. 1. Introduction The -comletion of the classifying sectrum of a finite grou is determined by the data of the induced fusion system on a Sylow -subgrou. That is, if G is a finite grou, S G is a Sylow -subgrou and F G is the fusion system on S determined by G, then there is an equivalence of sectra (Σ BG) Σ BF G, where BF G is a kind of classifying sace associated to the fusion system (see Section 2.8). The solution to the Segal conjecture rovides an algebraic descrition of the homotoy classes of mas between susension sectra of finite grous in terms of Burnside modules. In [Rag], a Burnside module between saturated fusions systems is defined. It is a submodule of the -comlete Burnside module between the Sylow -subgrous that is characterized in terms of the fusion data. It is shown that this submodule catures the stable homotoy classes of mas between the -comletions of susension sectra of finite grous. The - comletion functor induces a natural ma of abelian grous [Σ + BG, Σ + BH] [(Σ + BG), (Σ + BH) ]. In this aer, we give an algebraic descrition of this ma in terms of fusion data. Let G and H be finite grous. The roof of the Segal conjecture establishes a canonical natural isomorhism A(G, H) I G = [Σ + BG, Σ + BH] between the comletion of the Burnside module of finite (G, H)-bisets with free H-action at the augmentation ideal of the Burnside ring A(G) and the stable homotoy classes of mas between BG and BH. Fix a rime and Sylow -subgrous S and T of G and H resectively. Let F G and F H be the fusion systems on the fixed Sylow -subgrous determined by G and H. It follows from [BLO2] that there are canonical (indeendent of the choice of Sylow -subgrou) equivalences of sectra (Σ BG) Σ BF G and (Σ + BG) Σ BF G (S 0 ) (Σ + BF G ). The first author is suorted by the Danish Council for Indeendent Research s Saere Aude rogram (DFF ). 1

2 2 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON The Burnside module for the fusion systems F G and F H, as defined in [Rag], is the submodule AF (F G, F H ) A(S, T ) consisting of fusion stable (S, T )-bisets. Stability is defined entirely in terms of the fusion data. The restriction of a (G, H)-biset G X H to the Sylow -subgrous S and T induces a ma of Burnside modules A(G, H) A(S, T ) A(S, T ) sending G X H to S X T. This ma lands inside the stable elements: A(G, H) A(S, T ) AF (F G, F H ). We will write FG X FH for S X T viewed as an element of AF (F G, F H ). Corollary 9.4 of [RS] essentially roduces an isomorhism Using the -comletion functor we can form the comosite AF (F G, F H ) = [(Σ + BG), (Σ + BH) ]. ( ) : [Σ + BG, Σ + BH] [(Σ + BG), (Σ + BH) ] c: A(G, H) A(G, H) I G = [Σ + BG, Σ + BH] ( ) [(Σ + BG), (Σ + BH) ] = AF (F G, F H ). It is natural to ask for a comletely algebraic descrition of this ma in terms of bisets. This is not just the restriction ma, an extra ingredient is needed. Let T H T be the underlying set of H acted on the left and right by T. Since this is the restriction of H H H, it is stable so we may consider it as an element FH H FH AF (F H, F H ). It is invertible as H/T is rime to. Theorem 1.1. (Proosition 3.9) The comletion ma is given by c: A(G, H) AF (F G, F H ) c( G X H ) = ( FH H FH ) 1 ( FG X FH ) = FG X T H 1 F H. Thus we have a commutative diagram [Σ + BG, Σ + BH] A(G, H) ( ) [(Σ + BG), (Σ + BH) ] = c AF (F G, F H ), where c is given by the formula in the theorem above. Along the way to roving Theorem 1.1, we review the theory of Burnside modules, sectra, fusion systems, Burnside modules for fusion systems, and -comletion as well as roving a few folklore results. We rove that the susension sectrum of the -comletion of the classifying sace of a finite grou is the same as the -comletion of the classifying sectrum Σ (BG ) (Σ BG).

3 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 3 We also show that the -comletion ma induces an isomorhism [Σ + BG, Σ + BH] = [(Σ + BG), (Σ + BH) ] and give exlicit formulas for ( FH H FH ) 1 that aid comutation. In the final section, Section 4, we describe several categories and functors relevant for studying -comletion of classifying saces. Let G be the category of finite grous, G syl the category of finite grous with a chosen Sylow -subgrou, and F the category of saturated fusion systems. Let AG, AG syl, and AF be the corresonding Burnside categories. Finally, let Ho(S) and Ho(S ) be the homotoy categories of sectra and -comlete sectra. The functor c from Theorem 1.1 above fits together with -comletion of sectra and the functor F : G syl F associating a fusion system to each grou to give the commutative diagram G A AG α Ho(S) U AU G syl A syl AG syl ( ) F F c A fus AF β Ho(S ). Acknowledgments. All three authors would like to thank the Max Planck Institute for Mathematics and the Hausdorff Research Institute for Mathematics for their hositality and suort. The second and third author would like to thank SFB 1085 for its suort. The first author would like to thank his working grou at HIM for being atient listeners to discussions about bisets, I-adic toologies, and -comletions. 2. Preliminaries We recall definitions and results that are relevant to this aer. In articular, we review Burnside modules, sectra, fusion systems, and notions of -comletion Burnside modules. Definition 2.2. Let AG be the Burnside category of finite grous. The objects are finite grous. The morhism set between two grous G and H, AG(G, H), is the Grothendieck grou of isomorhism classes of finite (G, H)-bisets with free H-action and disjoint union as addition. We will refer to the elements in AG(G, H) as virtual bisets. Given a third grou K, the comosition ma AG(H, K) AG(G, H) AG(G, K) is induced by the ma sending an (H, K)-biset Y and a (G, H)-biset X to the coequalizer X H Y. The comosition ma is bilinear. There is a canonical basis of AG(G, H) as a Z-module given by the isomorhism classes of transitive (G, H)-bisets. These bisets are of the form G ϕ K H = (G H)/ (gk, h) (g, ϕ(k)h), where K G is a subgrou of G (taken u to conjugacy in G) and ϕ: K H is a grou homomorhism (taken u to conjugacy in G and H). We will denote these (G, H)-bisets by

4 4 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON [K, ϕ] H G or just [K, ϕ] when G and H are clear from context. It is also common to denote a virtual biset X AG(G, H) as G X H when G and H are not clear from context. There is also an unointed version of the category AG: Definition 2.3. Let KG be the category with objects finite grous and morhism sets given by ɛ KG(G, H) = ker(ag(g, H) AG(G, e)), where ɛ( G X H ) = G (X/H) e. Let A(G) = AG(G, e) be the Burnside ring of G. The multilicative structure comes from the roduct of left G-sets. This may be identified with the (commutative) subring A char (G) AG(G, G) sanned by the (G, G)-bisets of the form G K G = [K, i K ] (known as the semicharacteristic (G, G)-bisets), where K acts through the inclusion i K : K G on both sides. The identification of A char (G) with A(G) is given by the comosite A char (G) AG(G, G) ɛ AG(G, e). The inverse isomorhism sends G/H A(G) to G/H G = G H G A char (G), where the left action of G on G/H G is diagonal. Since AG(G, H) is a left AG(G, G)-module, it is also a left A(G)-module. Finally, let I G A(G) be the kernel of the augmentation A(G) Z sending a G-set X to its cardinality. Lemma 2.4 ([MM]). If G is a -grou of order n, then I n+1 G I G. Consequently, the I G - adic and -adic toologies on KG(G, H) coincide and the (+I G )-adic and -adic toologies on AG(G, H) coincide. Proof. This is a consequence of Lemma 5 in [MM]. In articular, the inclusion of ideals I n+1 G I G is art of the roof of that lemma Sectra. Definition 2.6. Let Ho(S) be the homotoy category of sectra. For a ointed sace X, let Σ X be the susension sectrum of X. Given another ointed sace Y, we let [Σ X, Σ Y ] be the abelian grou of homotoy classes of stable mas. For X unointed, let Σ + X be the susension sectrum of X with a disjoint baseoint. When G is a finite grou we will write Σ + BG for the susension sectrum of the classifying sace BG with a disjoint baseoint and Σ BG for the susension sectrum of BG using Be BG as the baseoint. There is a canonical ma sending [K, ϕ] H G to the comosite AG(G, H) [Σ + BG, Σ + BH] Σ + BG Tr Σ + BK Σ + Bϕ Σ + BH, where Tr is the transfer. This ma was intensely studied over several decades culminating in the following theorem. Theorem 2.7 ([Car], [AGM], [LMM]). There are canonical isomorhisms and [Σ + BG, S 0 ] = A(G) I G [Σ + BG, Σ + BH] = AG(G, H) I G.

5 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 5 There are several corollaries to this theorem (see [May], for instance). They include the following canonical isomorhisms and for S a -grou, due to Lemma 2.4, and [Σ BG, Σ BH] = KG(G, H) I G [Σ BS, Σ BH] = KG(S, H) [Σ + BS, Σ + BH] = AG(S, H). Since Σ + BG Σ BG S 0 Σ BG S 0, a ma Σ + BG Σ + BH is determined by four mas between the summands. However, since [S 0, Σ BH] = 0, only three of the mas are relevant. On the algebraic side, this corresonds to a slitting of the Burnside module AG(G, H) I G into three summands. This decomosition is induced by the orthogonal idemotents [G, 0], ([G, i G ] [G, 0]) AG(G, G) and [H, 0], ([H, i H ] [H, 0]) AG(H, H), where 0 is the ma that factors through the trivial subgrou. By multilying on the left or right aroriately, these two airs of idemotents give a slitting of AG(G, H) I G into four summands KG(G, H) I G I G Z 0, one of which is always zero. Because of this, most of the results of this aer can be converted into unointed results by multilying on the right with ([H, i H ] [H, 0]) Fusion systems. We recall the very basics of the definition of a saturated fusion system. For additional details see [Ree2, Section 2], [RS, Section 2] or [AKO, Part I]. We also discuss the construction of the classifying sectrum of a fusion system. Definition 2.9. A fusion system on a finite -grou S is a category F with the subgrous of S as objects and where the morhisms F(P, Q) for P, Q S satisfy (i) Every morhism ϕ F(P, Q) is an injective grou homomorhism ϕ: P Q. (ii) Every ma ϕ: P Q induced by conjugation in S is in F(P, Q). (iii) Every ma ϕ F(P, Q) factors as P ϕ ϕ(p ) Q in F and the inverse isomorhism ϕ 1 : ϕ(p ) P is also in F. A saturated fusion system satisfies some additional axioms that we will not go through as they lay no direct role in this aer. Given fusion systems F 1 and F 2 on -grous S 1 and S 2, resectively, a grou homomorhism ϕ: S 1 S 2 is said to be fusion reserving if whenever ψ : P Q is a ma in F 1, there is a corresonding ma ρ: ϕ(p ) ϕ(q) in F 2 such that ϕ Q ψ = ρ ϕ P. Note that each such ρ is unique if it exists. Examle Whenever G is a finite grou with Sylow -subgrou S, we associate a fusion system on S denoted F G. The mas in F G (P, Q) for subgrous P, Q S are recisely the homomorhisms P Q induced by conjugation in G. The fusion system F G associated to a grou at a rime is always saturated. Every saturated fusion system F has a classifying sectrum originally constructed by Broto-Levi-Oliver in [BLO2, Section 5]. The most direct way of constructing this sectrum, due to Ragnarsson [Rag], is as the maing telescoe Σ + BF = colim(σ + BS ω F Σ + BS ω F...),

6 6 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON where ω F : Σ + BS Σ + BS is the characteristic idemotent of F (see [Rag, Definition 4.3]). By construction Σ + BF is a wedge summand of Σ + BS. The transfer ma t: Σ + BF Σ + BS is the inclusion of Σ + BF as a summand and the inclusion ma r : Σ + BS Σ + BF is the rojection on Σ + BF. Thus r t = 1 and t r = ω F. As remarked in Section 5 of [BLO2], the sectrum Σ + BF constructed this way is in fact the susension sectrum for the classifying sace BF defined in [BLO2, Che]. One way to see this is to note that H (BF, F ) coincides with ω F H (BS, F ) as the F-stable elements, and by an argument similar to Proosition 2.14 later on, the susension sectrum of BF is HF -local Burnside modules for fusion systems. Fix a rime. Definition Let AF be the Burnside category of saturated fusion systems. The objects in this category are saturated fusion systems (F, S) over finite -grous. Let F 1 and F 2 be saturated fusion systems on -grous S 1 and S 2. The morhisms in AF between (F 1, S 1 ) and (F 2, S 2 ) are the elements of the Burnside module AF (F 1, F 2 ) AF (S 1, S 2 ) = AG(S 1, S 2 ). This is the submodule of the -comlete Burnside module AG(S 1, S 2 ) consisting of left F 1 -stable and right F 2 -stable elements. Stability may be defined in two ways. We say that an element X AF (S 1, S 2 ) is left F 1 -stable if ω 1 X = X and right F 2 -stable if X ω 2 = X, where ω 1 and ω 2 are the characteristic idemotents of F 1 and F 2. Algebraically, the definition is longer but more elementary. An (S 1, S 2 )-biset X is left F 1 -stable if for all airs of subgrous P, Q S 1 and any isomorhism ϕ: P = F1 Q in F 1 the (P, S 1 )-sets P X S1 and ϕ P X S 1 are isomorhic, where is the biset induced by restriction along ϕ P X S 1 P ϕ Q S 1. Right stability is defined similarly. The Burnside module AF (F 1, F 2 ) is the -comletion of the Grothendieck grou of left F 1 -stable right F 2 -stable (S 1, S 2 )-bisets ([Ree1, 4.4]). For short, we will call such left F 1 -stable right F 2 -stable elements stable. It is worth noting that there is an inclusion AG(S 1, S 2 ) AF (S 1, S 2 ) so we may view any (S 1, S 2 )-biset as an element in AF (S 1, S 2 ). In fact, AF (S 1, S 2 ) is the free Z -module on bisets of the form [K, ϕ] S2 S 1. Similarly, AF (F 1, F 2 ) is a free Z -module on virtual bisets of the form [K, ϕ] F2 F 1 = ω F1 [K, ϕ] S2 S 1 ω F2, where K and ϕ are taken u to conjugacy in F 1 and F 2 ([Rag, 5.2]). In order to clarify that a stable (S 1, S 2 )-biset X is being viewed in AF (F 1, F 2 ) we will write F1 X F2. Given a third saturated fusion system F 3 on S 3 and bisets F1 X F2 and F2 Y F3, we will denote the comosite biset by F 1 X S2 Y F3 = ( F1 X F2 ) S2 ( F2 Y F3 ).

7 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 7 Given finite grous G and H with Sylow subgrous S and T and a (G, H)-biset X, we may restrict the G-action to S and the H-action to T to get an (S, T )-biset S X T. Let F G be the fusion system associated to G on S and and F H the fusion system associated to H on T. The restricted biset S X T is always a stable biset and so we may further consider it as an (F G, F H )-biset F G X FH. We turn our attention to Burnside rings for saturated fusion systems. Recall that the comosite A char (G) AG(G, G) AG(G, e) = A(G) of Section 2.1 is an isomorhism and identifies the Burnside ring A(G) with the subring of AG(G, G) on the semicharacteristic (G, G)-bisets. In the same way, there are two versions of the Burnside ring associated to a fusion system F on a -grou S. The first, denoted A(F), is the subring of F-stable elements of A(S). The second is the subring of A char (F) AF (F, F) AF (S, S) consisting of F-semicharacteristic bisets. This is the Z -submodule sanned by the basis elements of the form [K, i K ] F F. The identity element in Achar (F) is the characteristic idemotent. The units of A char (F) are usually referred to as the F-characteristic elements, and each of them contains enough information to reconstruct F (see [RS, Theorem 5.9]). The commutative rings A(F) and A char (F) may be canonically identified after -comletion, but it is useful to distinguish between the two. The (non-multilicative) ma ɛ: AG(S, S) A(S) induces a ma A char (F) AF (F, F) A(F) which is a ring isomorhism by Theorem D in [Ree2]. Let I F be the kernel of the augmentation ma I F = ker(a(f) A(S) Z). The ring A(F) is clearly -comlete. It is also comlete with resect to the maximal ideal () + I F and these ideals give the same toology. This follows immediately from the fact that the ideals () and () + I S give the same toology on A(S) (Lemma 2.4) comletion. Let E be a sectrum. There is a -comletion functor on sectra equied with a canonical transformation E E. This functor is given by Bousfield localization at the Moore sectrum MZ/. When E is connective, for instance if E is the classifying sectrum of a finite grou, then E is also the localization of E at HF. There is a natural equivalence (Σ + BG) (Σ BG) (S 0 ). The arithmetic fracture square immediately imlies that Σ BG If S is a -grou, then Σ BS (Σ BS) so (Σ BG). (Σ + BS) Σ BS (S 0 ). When the rime is clear from context and X is a sace, we will write ˆΣ + X = (Σ + X)

8 8 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON for the -comletion of the susension sectrum with a disjoint baseoint. Let S G be a Sylow -subgrou. Since ˆΣ + BG is a retract of ˆΣ + BS and because mas to -comlete sectra factor through the -comletion, the restriction along the inclusion induces an injection [Σ + BG, (S 0 ) ] [Σ + BS, (S 0 ) ]. There are several notions of -comletion for saces. These were develoed in [Bou1], [BK], [Sul1], and [Sul2]. For a sace such as BG, these notions all agree and there is a simle relationshi between the stable -comletion and the unstable -comletion. Since it is difficult to find a roof of this fact in the literature, we rovide a comlete roof. Proosition Let G be a finite grou. There is a canonical equivalence Σ (BG ) (Σ BG). Proof. It follows from [BK, VII.4.3] that the unstable homotoy grous π (BG ) are all finite -grous. This imlies that the reduced integral homology grous HZ (BG ) are all finite -grous: Let BG be the universal cover. A Serre class argument with the Serre sectral sequence associated to the fibration BG BG K(π 1 (BG ), 1) reduces this roblem to the grou homology of π 1 (BG ) with coefficients in the integral homology of the fiber. The result follows from the fact that the reduced homology grous of the fiber are -grous and that the integral grou homology grous of π 1 (BG ) are -grous. Both of these are classical (see [DK, Chater 10] for the fiber, for instance). This imlies that the stable homotoy grous π (Σ (BG )) are all finite -grous: This is a Serre class argument with the (convergent) Atiyah-Hirzebruch sectral sequence. This imlies that the sectrum Σ (BG ) is -comlete: As Σ (BG ) is connective, it suffices to rove that the sectrum is HF -local. Let X be an HF -acyclic sectrum. Let Y i be the ith stage in the Postnikov tower for Σ (BG ) so that Σ (BG ) lim Y i and Σ (BG ) X lim Yi X. We would like to show that this sectrum is zero. Let K i be the fiber of the ma Y i Y i 1. By induction, it suffices to rove the Ki X 0. There is an equivalence K i Σ k HA for a finite abelian -grou grou A. Thus it suffices to show that (Σ k HZ/ l ) X 0 for all l. By induction on the fiber sequence Σ k HF Σ k HZ/ l Σ k HZ/ l 1 it suffices to rove that (Σ k HF ) X 0. This is the sectrum of HF -module mas Mod HF (HF X, Σ k HF ). By assumtion HF X. This imlies that the canonical ma factors through Σ BG Σ (BG ) (Σ BG) Σ (BG ) which is an HF -homology equivalence between HF -local sectra and thus is an equivalence.

9 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 9 When restricted to the homotoy category of classifying sectra of finite grous, the -comletion functor has a simle descrition. We have not found this fact in the literature. Proosition The -comletion functor on sectra induces an isomorhism Proof. We have slittings [Σ + BG, Σ + BH] = [ˆΣ + BG, ˆΣ + BH]. [Σ + BG, Σ + BH] = [S 0, S 0 ] [Σ BG, S 0 ] [Σ + BG, Σ BH]. and Σ BG l (Σ BG) l and Σ BH l (Σ BH) l, where the wedges are over rimes dividing the order of the grou. Since [X, (Σ BH) l ] is l-comlete for finite tye X and a rime l and thus algebraically l-comlete ([Bou2, 2.5]), the (algebraic) -comletion of the abelian grou [Σ + BG, Σ BH] is [Σ + BG, ˆΣ BH]. The algebraic -comletion of [S 0, S 0 ] is clearly Z = [S 0, (S 0 ) ]. Finally, we must deal with [Σ BG, S 0 ]. However, since Σ BG is connected, mas from Σ BG to S 0 factor through the connected cover of S 0, S 0. The connected cover of S 0 has trivial rational cohomology, so the arithmetic fracture square gives a slitting S 0 l ( S 0 ) l l ( S 0 ) l. The second equivalence is a consequence of the fact that π i S 0 is finite above degree zero. Thus the grou [Σ BG, S 0 ] aears to be an infinite roduct, however the l-comletion of Σ BG is contractible for l not dividing the order of G. Thus the roduct [Σ BG, ( S 0 ) l ] l has a finite number of non-zero factors. The -comletion of this roduct is just the factor corresonding to the rime. 3. A formula for the -comletion functor Fix a rime. We give an exlicit formula for the -comletion functor from virtual (G, H)-bisets to -comlete sectra sending a biset X : Σ + BG Σ + BH to the -comletion X : ˆΣ + BG ˆΣ + BH.

10 10 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON 3.1. Further results on Burnside modules for fusion systems. We begin with a result describing how Burnside modules for fusion systems relate to the stable homotoy category. Let G and H be finite grous and let S G and T H be fixed Sylow -subgrous. Let F G and F H be the fusion systems on S and T determined by G and H. Recall that there are natural forgetful mas AG(G, H) AG(S, T ) and AF (F G, F H ) AF (S, T ). In the stable homotoy category the analogous mas are the ma [Σ + BG, Σ + BH] [Σ + BS, Σ + BT ] given by comosing with the inclusion from Σ + BS and the transfer to Σ + BT and the ma [ˆΣ + BF G, ˆΣ + BF H ] [ˆΣ + BS, ˆΣ + BT ] given by recomosing with r and ostcomosing with t. We may use this to construct a ma [Σ + BG, Σ + BH] [ˆΣ + BF G, ˆΣ + BF H ] by sending X to the -comletion of the comosite Σ + BF G t Σ + BS S G G Σ + BG X Σ + BH H H T Σ + BT r Σ + BF H. Proosition 3.2. Let F 1 and F 2 be saturated fusion systems on the -grous S 1 and S 2. There is a canonical isomorhism AF (F 1, F 2 ) = [ˆΣ + BF 1, ˆΣ + BF 2 ]. Proof. The abelian grous AF (S 1, S 2 ) and [ˆΣ + BS 1, ˆΣ + BS 2 ] both have canonical idemotent endomorhisms given by recomosing and ostcomosing with the characteristic idemotents associated to F 1 and F 2. The algebraic characteristic idemotent mas to the sectral characteristic idemotent by definition. This comatibility ensures that the images of these endomorhisms ma to each other under the canonical isomorhism of Proosition 2.15 AF (S 1, S 2 ) = [ˆΣ + BS 1, ˆΣ + BS 2 ]. The images of these endomorhisms are recisely the (F 1, F 2 )-stable bisets and the homotoy classes [ˆΣ + BF 1, ˆΣ + BF 2 ]. Since the retract of an isomorhism is an isomorhism, we have constructed a canonical isomorhism AF (F 1, F 2 ) = [ˆΣ + BF 1, ˆΣ + BF 2 ]. Proosition 3.3. Let G and H be finite grous and let S G and T H be Sylow -subgrous. Let F G and F H be the fusion systems on S and T determined by G and H.

11 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 11 There is a commutative diagram AG(G, H) [Σ + BG, Σ + BH] AG(S, T ) [Σ + BS, Σ + BT ] AF (S, T ) = [ˆΣ + BS, ˆΣ + BT ] AF (F G, F H ) = [ˆΣ + BF G, ˆΣ + BF H ]. Proof. The to center square commutes by naturality of Theorem 2.7. The middle center square commutes by the corollaries to Theorem 2.7. The bottom middle square commutes by the discussion in the roof of Proosition 3.2. The left art of the diagram commutes as the restriction of a (G, H)-biset is bistable. Note that the ma [ˆΣ + BF G, ˆΣ + BF H ] [ˆΣ + BS, ˆΣ + BT ] is given by the formula f tfr. The right art of the diagram commutes as the -comletion of the comosite Σ + BF G t Σ + BS S G G Σ + BG X Σ + BH H H T Σ + BT mas to the -comletion of the comosite r Σ + BF H Σ + BS ω F G Σ + BS S G G Σ + BG X Σ + BH H H T Σ + BT ω F H Σ + BT and stability imlies that this is equal to the -comletion of Σ + BS S G G Σ + BG X Σ + BH H H T Σ + BT. Proosition 3.3 gives an interretation of the biset construction GX H FG X FH in terms of sectra. It is the -comletion of the comosite Σ + BF G t Σ + BS S G G Σ + BG X Σ + BH H H T Σ + BT r Σ + BF H. Now we focus on the relationshi between ˆΣ + BF G and ˆΣ + BG. Alying -comletion to the mas t: Σ + BF G Σ + BS and S G G : Σ + BS Σ + BG gives us the commutative diagram (1) Σ t + BF G Σ + BS S G G Σ + BG ˆΣ + BF G ˆΣ + BS a G ˆΣ + BG. The ma a G is the comosite of the bottom arrows.

12 12 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON Proosition 3.4. ([CE, XII.10.1], [BLO2]) The ma is an equivalence. a G : ˆΣ + BF G ˆΣ + BG Note that a G is canonical and natural in mas of finite grous. We will use this equivalence to identify these sectra with each other. Let S G be a Sylow -subgrou and define J G = ker(a(g) A(S)). Note that J G is indeendent of the choice of Sylow -subgrou. Corollary 3.5. Let G and H be finite grous and let J G A(G) be the ideal defined above. There are canonical isomorhisms natural in G and H. AG(G, H) +I G = Z AG(G, H)/J G AG(G, H) = AF (F G, F H ) Proof. Since A(G) +I G = Z A(G)/J G the first isomorhism follows from the fact that the comletion is given by base change. Naturality of this isomorhism follows from the fact that the image of a Sylow -subgrou under a grou homomorhism is contained in a Sylow -subgrou. To see the other isomorhism, recall that Theorem 2.7 gives an isomorhism AG(G, H) I G = [Σ + BG, Σ + BH]. In view of Proosition 3.4, it suffices to show that the -comletion functor [Σ + BG, Σ + BH] [ˆΣ + BG, ˆΣ + BH] is given algebraically by -comletion. This is Proosition A formula for -comletion. Now consider the ma S G S : Σ + BS Σ + BS, which is the comosite of the inclusion and transfer along S G. This element is F G -semicharacteristic; it is not S-semicharacteristic as it is the comosite SG S = [S, id S ] S G [S, i S ] G S and various conjugations with resect to elements of G not in S show u in the resulting sum. Lemma 3.7. The element is a unit. F G G FG AF (F G, F G ) = [ˆΣ + BF G, ˆΣ + BF G ] Proof. Since FG G FG is F G -semicharacteristic it is in the image of the inclusion Recall that the comosite i: A char (F G ) AF (F G, F G ). A char (F G ) AF (F G, F G ) A(F G ) is an isomorhism of commutative rings even though the second ma is not a ring ma. The image of FG G FG in A(F G ) is G/S viewed as an F G -stable set. Since G/S is corime to, this rojects onto a unit in F under the canonical ma A(F G ) Z F.

13 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 13 Since A(F G ) is comlete local with maximal ideal + I FG, G/S is a unit, but now this imlies that FG G FG is a unit. Recall the equivalence of Proosition 3.4, we now give an exlicit descrition of the inverse to a G. Consider the following diagram (2) Σ + BG G G S Σ + BS r Σ + BF G ˆΣ + BG ˆΣ + BS ˆΣ + BF G b G ˆΣ + BF G. ( FG G FG ) 1 The ma G G S is the transfer from Σ + BG to Σ + BS. The second row is the -comletion of the first row. The ma ( FG G FG ) 1 exists by Lemma 3.7, which deends on the fact that we are in the category of -comlete sectra. The ma b G is the comosite. Lemma 3.8. The ma is the inverse to a G. b G : ˆΣ + BG ˆΣ + BF G. Proof. Put Diagram 1 from age 11 to the left of Diagram 2. Note that the image of G G G along the ma AG(G, G) AF (F G, F G ) of Proosition 3.3 is FG G FG, which we then ostcomose with ( FG G FG ) 1. Using b G, we may relace the target of the canonical ma to the -comletion by ˆΣ + BF G. Let Σ + BG ˆΣ + BG c G : Σ + BG ˆΣ + BG b G ˆΣ + BF G be the comosite; it is naturally equivalent to the -comletion ma. Diagram 2 gives a kind of formula for c G. It is the the transfer G G S, viewed as a ma landing in Σ + BF G, ostcomosed with the -comletion ma Σ + BF G ˆΣ + BF G followed by the equivalence ( FG G FG ) 1 : c G : Σ + BG G G S Σ + BS r Σ + BF G ˆΣ + BF G Using the equivalences a G and b H, we may construct a ma ( ): [Σ + BG, Σ + BH] [ˆΣ + BF G, ˆΣ + BF H ] ( FG G FG ) 1 ˆΣ + BF G. by sending to the comosite f : Σ + BG Σ + BH f : ˆΣ + BF G a G ˆΣ + BG f ˆΣ + BH b H ˆΣ + BF H. By Proosition 3.2, this is an element in AF (F G, F H ). We will give a simle formula for f when f comes from a virtual (G, H)-biset. In fact, we may recomose ( ) with the

14 14 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON canonical ma A(G, H) [Σ + BG, Σ + BH]. By abuse of notation, we will also call this ( ). Proosition 3.9. Let G and H be finite grous, and let T H be the Sylow -subgrou on which F H is defined. The ma sends a virtual (G, H)-biset G X H to F G XFH AG(G, H) ( ) AF (F G, F H ) = FG X T H 1 F H = ( FG X FH ) T ( FH H FH ) 1. In other words, there is a commutative diagram in the stable homotoy category Σ + BG X Σ + BH c G ˆΣ + BF G F G X T H 1 c H F H ˆΣ + BF H. Proof. Putting together Diagram 1 and the definition of c H, we have a commutative diagram Σ + BF G Σ + BS Σ + BG X Σ + BH ˆΣ + BF G a G ˆΣ + BG X ˆΣ + BH b G ˆΣ + BF H. c H The vertical arrows are all the canonical mas to the -comletion. Note that X is the is the comosite of the arrows in the bottom row of the diagram. Also, we could add c G : Σ + BG ˆΣ + BF G diagonally in the left hand square and the diagram would still commute. The comosite along the to is recisely giving us the desired formula. F G X T H 1 F H Remark The formula of Proosition 3.9 also makes sense on elements in A(G, H) I G and the roof is identical once one feels comfortable referring to elements in the comletion as virtual bisets. This result allows us to give exlicit formulas for the -comletion functor. It is often useful to have formulas more exlicit than ( FG G FG ) 1. We will give two further ways of understanding this element. One as an infinite series and the other as a certain limit. The following formulas for calculating ( FG G FG ) 1 are based on similar calculations in [Rag]. Proosition Let X = FG G FG. Inside AF (F G, F G ) we have the equalities X 1 = X 2 i 0(1 X 1 ) i = lim n X( 1)n 1. Proof. Since X is F G -characteristic, it suffices to rove this inside A(F G ) = A char (F G ) AF (F G, F G ). This ring is comlete local with maximal ideal m = + I FG.

15 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 15 Since X is invertible, it is enough to show that lim n X( 1)n = X lim 1 n X( 1)n = 1. But since A(F G ) /m = F, we have that X 1 = 1 mod m. Thus it is enough to show that if Y = 1 mod m, then lim n Y n = 1. Indeed, we will rove by induction that Y n 1 m n+1. The case n = 0 follows by assumtion. Assume that Y n 1 m n+1. We have but 1 Y n+1 1 = (Y n 1)( Y ni ), i=0 1 1 Y ni = 1 ni = = 0 mod m i=0 i=0 and (Y n 1) m n+1 by assumtion, so (Y n+1 1) m n+2. For the other equality, note that the sum is the geometric series for 1/X 1 and that the summand live in higher and higher owers of m. Remark The formulas for X 1 in the revious roosition are true much more generally. For instance, it suffices that R is a (not necessarily commutative) Z -algebra with a two-sided maximal ideal m such that R/m = F and R is finitely generated as a Z -module. Corollary The idemotent in KG(G, G) I G that slits off (Σ BG) as a summand of Σ BG can be written as lim n ([S, i S] G G [S, 0] G G) ( 1)n. The biset [S, i S ] G G [S, 0]G G = (G S G) (G/S e G) is just the transfer from Σ BG to Σ BS followed by the inclusion back to Σ BG. Proof. Recall the discussion following Proosition 2.7, that in order to get the unointed version Σ BG Σ BH of any ma Σ + BG Σ + BH, we just have to ostcomose with the idemotent [H, i H ] [H, 0] AG(H, H). This is because, for any virtual (G, H)-biset X, we have (3) ([G, i G ] [G, 0]) G X H ([H, i H ] [H, 0]) = X H ([H, i H ] [H, 0]). The ma c G : Σ + BG ˆΣ + BF G has an unointed version c G : Σ BG Σ BF G which we get by ostcomosing with the idemotent ω F [S, 0] S S = ω F S ([S, i S ] S S [S, 0]S S ) AF (F, F). The ma c G is reresented by the comosition Σ BG G G S Σ BS or equivalently as the element Similarly the ma s: Σ BF G SG G G ([G, i G ] [G, 0]). r Σ BF G ( FG G FG ) 1 Σ BF G GG S S ( FG G FG ) 1 S (ω F [S, 0]). t Σ BS i S Σ BG is reresented by the virtual biset

16 16 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON We see that s is a section to c G since c G s is reresented by the comosite SG G G ([G, i G ] [G, 0]) G G G S S ( FG G FG ) 1 S (ω F [S, 0]) = S G G G G G S S ( FG G FG ) 1 S (ω F [S, 0]) = S G S S ( FG G FG ) 1 S (ω F [S, 0]) = (ω F [S, 0]) and ω F [S, 0] is the identity ma on Σ BF G. Note that we can remove ([G, i G ] [G, 0]) in the middle since we multily by (ω F [S, 0]) at the end anyway. The comosition c G s ends with the equivalence b G : (Σ BG) Σ BF G. If we instead lace b G at the beginning, we see that (Σ BG ) b G (Σ BF G ) s Σ BG (Σ BG) is also the identity. From this we conclude that s c G : Σ BG Σ BG is an idemotent with image the -comletion (Σ BG). We now lug in the limit formula for ( FG G FG ) 1 from Proosition 3.11 and see that the idemotent s c G has the form GG S S ( FG G FG ) 1 S (ω F [S, 0]) S S G G G ([G, i G ] [G, 0]) = G G S S ( FG G FG ) 1 S S G G G ([G, i G ] [G, 0]) ( ) = G G S S lim ( SG S ) ( 1)n 1 S S G G G ([G, i G ] [G, 0]) n ( = lim (G S G) ( 1)n) G ([G, i G ] [G, 0]) n = lim n (G S G [S, 0] G G) ( 1)n. The last equality holds because (3) tells us that we can act with the idemotent [G, i G ] [G, 0] on every factor in a long comosition, and (G S G) G ([G, i G ] [G, 0]) = G S G [S, 0] G G. Examle As an examle of how the different formulas of this aer lay together with the I G -adic toology, we will erform a sanity check. We will check that the idemotents of Corollary 3.13 at each rime actually add u to give back the identity on Σ BG. Let S be a Sylow -subgrou of G, and let ω denote the idemotent ω := lim n ([S, i S ] G G [S, 0] G G) ( 1)n that slits off (Σ BG) from Σ BG. We will confirm that [G, i G ] [G, 0] = ω in the endomorhism ring KG(G, G) I G of Σ BG. First note that we may write (4) 1 = a G S, a linear combination of integers for some choice of integers a. Next let Z := which is an element of I G ([G, i G ] [G, 0]). a ( G S ([G, i G] [G, 0]) ([S, i S ] [S, 0])),

17 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 17 We now claim that Z G (([G, i G ] [G, 0]) ω ) = ([G, i G ] [G, 0]) ω. To show this we need the following two calculations: The first is the fact that ([S, i S ] [S, 0]) G ([S q, i Sq ] [S q, 0]) = 0 whenever q, because any conjugates of S and S q in G always intersect trivially. Hence ([S, i S ] [S, 0]) G ω q = 0 as ω q is formed by iterating [S q, i Sq ] [S q, 0]. The second calculation we need is that ([S, i S ] [S, 0]) G ω = lim n ([S, i S ] [S, 0]) G ([S, i S ] [S, 0]) ( 1)n = ([S, i S ] S G [S, 0] S G ) S ( lim n ( S G S ( S G/S e S )) ( 1)n) S ([S, i S ] G S [S, 0] G S ) = ([S, i S ] S G [S, 0] S G ) S (ω FS (G) [S, 0] S S ) S ([S, i S ] G S [S, 0] G S ) = ([S, i S ] S G [S, 0] S G ) S ([S, i S ] G S [S, 0] G S ) by F S (G)-stability = [S, i S ] [S, 0]. This imlies that [S, i S ] G G [S, 0] G G is ω -stable (not surrisingly). Now we return to the roduct with Z: Z G ( ([G, i G ] [G, 0]) ω ) ( = ( = ( G )) ( a S ([G, i G] [G, 0]) ([S, i S ] [S, 0]) G ([G, i G ] [G, 0]) G ) ( a ([G, i G ] [G, 0]) ) ω S ( ) ( a ([S, i S ] [S, 0]) G ([G, i G ] [G, 0]) ) ω ω )

18 18 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON By Equation 4, this is equal to ( ([G, i G ] [G, 0]) ) ω ( ) ( a ([S, i S ] [S, 0]) G ([G, i G ] [G, 0]) ) ω ( = ([G, i G ] [G, 0]) ) ω ( ( )) a ([S, i S ] [S, 0]) G ([G, i G ] [G, 0]) ([S, i S ] [S, 0]) G w q = = ( ([G, i G ] [G, 0]) ( ([G, i G ] [G, 0]) = ([G, i G ] [G, 0]) ) ( ω ) ( ω ω. a ( ([S, i S ] [S, 0]) ([S, i S ] [S, 0]) G ω )) ) a 0 q Since Z is in I G ([G, i G ] [G, 0]), this shows that ([G, i G ] [G, 0]) ω is in IG k KG(G, G) for all k and therefore equal to 0 in the I G -adic comletion. Thus ([G, i G ] [G, 0]) = ω as we claimed. 4. Categories related to fusion systems We introduce several categories closely connected to the category of fusion systems and study some of the functors between them. We aly the formula for -comletion of the revious section to roduce a commutative diagram involving these categories. Definition 4.1. Let G be the category of finite grous and grou homomorhisms. Fix a rime. Definition 4.2. Let G syl be the category with objects airs (G, S), where G is a finite grou and S is a Sylow -subgrou of G. A morhism between two objects (G, S) and (H, T ) is a homomorhism f : G H such that f(s) T. Definition 4.3. Let F be the category of saturated fusion systems. The objects are saturated fusion systems (F, S) and a morhism from (F, S) to (G, T ) is a fusion reserving grou homomorhism S T. Recall that AG is the Burnside category of finite grous. Objects are finite grous and the morhism set between two grous G and H, AG(G, H), is the Grothendieck grou of finite (G, H)-bisets with a free H-action. Also recall that AF is the Burnside category of fusion systems. Definition 4.4. Let AG syl be the category with objects airs (G, S) where G is a finite grou and S is a Sylow -subgrou of G and with morhisms between two objects (G, S) and (H, T ) given by AG syl ((G, S), (H, T )) = AG(G, H). We will also make use of several categories coming from homotoy theory.

19 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 19 Definition 4.5. Let Ho(To G ) be the full subcategory of the homotoy category of saces on the classifying saces of finite grous. Let Ho(S) be the homotoy category of sectra and let Ho(S ) be the homotoy category of -comlete sectra. We describe the relationshis between these categories by defining several canonical functors. We roduce commutative diagrams involving the categories, alying the formula for -comletion from the revious section. Let A: G AG be the canonical functor from the category of grous to the Burnside category. It takes a grou homomorhism ϕ: G H to the (G, H)-biset [G, ϕ] H G, which is just ϕ G H H with G acting through ϕ on the left. The functor A is not faithful, conjugate mas are identified. It does factor through the full functor B( ) to Ho(To G ) and Ho(To G ) does ma faithfully into AG. Let U : G syl G be the forgetful functor, sending (G, S) to G. This functor is faithful. Given a ma ϕ: G H and a Sylow subgrou S G, ϕ(s) is contained in some Sylow subgrou of H and this rovides a lift of ϕ to G syl. Let A syl : G syl AG syl be defined just as the functor A. It takes a morhism ϕ: (G, S) (H, T ) to the biset [G, ϕ] H G. Note that any ma G H is H-conjugate to a ma sending S into T. Thus the image of G syl ((G, S), (H, T )) in AG syl ((G, S), (H, T )) = AG(G, H) is equal to the image of G(G, H) under the functor A. Let AU : AG syl AG be the forgetful functor. Note that this functor is an equivalence, it is fully faithful and surjective on objects. Let F : G syl F be the functor sending a air (G, S) to the induced fusion system F G on S. By construction, the morhisms in G syl restrict to fusion reserving mas between the chosen Sylow -subgrous. Let A fus : F AF be the functor that is the identity on objects and takes a ma of fusion systems (F, S) (G, T ) induced by a fusion reserving ma ϕ: S T to [S, ϕ] G F. In Proosition 4.7 below, we rove that this is a functor. Lemma 4.6. Let ϕ: S T be a fusion reserving ma between saturated fusion systems (F, S) and (G, T ). Then ω F S [S, ϕ] T ω G = [S, ϕ] T ω G. Proof. It is sufficient to show that X := [S, ϕ] T ω G is left F-stable. To see that X is F-stable we consider an arbitrary subgrou P S and ma ψ F(P, S) and rove that the restriction of X along ψ, ψ P X T, is isomorhic to P X T as virtual (P, T )-bisets. The restriction of [S, ϕ] T S along ψ : P S is just [P, ϕ ψ]t P. We therefore have ψ P X T = [P, ϕ ψ] T P T ω G. Since ψ is a ma in F, and since ϕ is assumed to be fusion reserving, this means that there is some ma ρ: ϕ(p ) T in G such that ϕ ψ(p ) ψ = ρ ϕ P. Finally, ω G absorbs mas in G, and thus ψ P X T = [P, ϕ ψ] T P T ω G = [P, ρ ϕ] T P T ω G = [P, ϕ] T P T ω G = P X T. Proosition 4.7. The oeration A fus described above is a functor. Proof. Suose we have two fusion reserving mas ψ : R S, ϕ: S T between saturated fusion systems (E, R), (F, S) and (G, T ). Alying Lemma 4.6 to ϕ, we easily confirm that

20 20 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON A fus reserves comosition: A fus (ϕ) A fus (ψ) = ω E R [R, ψ] S ω F S [S, ϕ] T ω G = ω E R [R, ψ] S [S, ϕ] T ω G = ω E R [R, ϕ ψ] T ω G = A fus (ϕ ψ). Let AF : AG syl AF be the functor taking (G, S) to F G and taking a virtual (G, H)- biset X to F G XFH as described in Proosition 3.9. This is the functor c from Theorem 1.1 of the introduction. Proosition 4.8. There is a commutative diagram of categories G A AG U G syl F F AU A syl AG syl AF A fus AF. Proof. The to square clearly commutes, so we need to rove that the bottom square commutes as well. Let ϕ: G H be a morhism in G syl from (G, S) to (H, T ), meaning that ϕ(s) T. The functor A syl takes ϕ to the biset [G, ϕ] H G. To understand what haens when we aly AF to [G, ϕ] H G, we first need to understand the restriction S([G, ϕ] H G ) T of the biset to the Sylow -subgrous. We can describe the restriction S ([G, ϕ] H G ) T as comosing with the inclusion biset S G G on the left and the transfer biset H H T on the right: S([G, ϕ] H G ) T = S G G [G, ϕ] H G H H T. Now S G G [G, ϕ] H G simly gives us the restriction of ϕ to the subgrou S, [S, ϕ S] H S. By assumtion ϕ S lands in T H, and therefore S([G, ϕ] H G ) T = [S, ϕ S ] H S H H T = [S, ϕ S ] T S T H H H T = [S, ϕ S ] T S T H T. The biset T H T is F H -stable and invertible inside AF (F H, F H ), and the functor AF alied to [G, ϕ] H G is by Proosition 3.9 equal to AF (A syl (ϕ)) = S ([G, ϕ] H G ) T ( FH H FH ) 1 = [S, ϕ S ] T S T H T ( FH H FH ) 1 = [S, ϕ S ] T S T ω FH = [S, ϕ S ] F H F G = A fus (F (ϕ)) The enultimate equality is due to Lemma 4.6 since the restriction ϕ S is a fusion reserving ma from F G to F H.

21 A FORMULA FOR -COMPLETION BY WAY OF THE SEGAL CONJECTURE 21 Let α: AG syl Ho(S) be the functor sending G to Σ + BG and sending [K, ϕ] H G to the comosite Σ + BG TrK G Σ + BK Σ + Bϕ Σ + BH, where Tr is the transfer. This functor is well-understood by the solution to the Segal conjecture. It is neither full nor faithful. Let β : AF Ho(S ) be the analogous functor for fusion systems. It sends the object F to Σ + BF and alies Proosition 3.2 to mas. The functor β is fully faithful. Let ( ) be the -comletion functor Ho(S) Ho(S ). Proosition 4.9. There is a commutative diagram AU AG α Ho(S) AG syl ( ) AF AF β Ho(S ) u to canonical natural equivalence and the formula for AF is given by Proosition 3.9. Proof. This is a consequence of Proosition 3.9. Let AG free (G, H) be the submodule of the Burnside module AG(G, H) generated by bisets which have a free action by both G and H. This submodule has a basis consisting of isomorhism classes of sets of the form [K, ϕ] H G, where ϕ is an injection. This includes the biset G H H = [G, i] H G = A syl(i), where G acts on H through an injection i: G H. Since we have a natural isomorhism sending ( ) o : AG free (G, H) = AG free (H, G) AG(H, G) GX H ( G X H ) o = H X G we find that elements of AG free (G, H) give rise to mas in AG not only from G to H but also from H to G. The image under ( ) o of the elements of the form [G, i] H G, where i is an injection, are referred to as the transfer mas. The same story makes sense for fusion reserving injections between two fusion systems. Let F 1 and F 2 be saturated fusion systems on -grous S 1 and S 2. A bifree (F 1, F 2 )-biset is a bifree bistable (S 1, S 2 )-biset. The isomorhism induces an isomorhism ( ) o : Z AG free (S 1, S 2 ) = Z AG free (S 2, S 1 ) ( ) o : AF free (F 1, F 2 ) = AF free (F 2, F 1 ). A transfer ma from F 2 to F 1 is the image of an element of the form [S 1, i] F2 F 1 where i is an injection of fusion systems, under the ma ( ) o. Since the functor F takes injection to injections, Proosition 4.8 imlies that AF ([G, i] H G ) = [S, F (i)] F H F G, = A fus (i),

22 22 S. P. REEH, T. M. SCHLANK, AND N. STAPLETON where S G is a Sylow -subgrou. Thus AF takes injections to injections. It is temting to assume that the functor AF will also reserve transfer mas. However, in general, (AF ( G H H )) o AF (( G H H ) o ). To see this, let G = e and let H to be a non-trivial grou of order rime to. Consider the element [e, i] H e AG(e, H), where i is the inclusion of the identity element. This element gives rise to two mas in AG the restriction e H H and the transfer H H e, and comosing those we get the element e H e = H AG(e, e) = Z. Since both e and H have trivial Sylow -subgrous, Proosition 4.8 imlies that and thus However, Since AF ( e H H ) = Id Fe, AF ( e H H ) = Id Fe (AF ( e H H )) o = Id Fe. AF ( H H e ) AF ( e H H ) = AF ( e H e ) = H AF (e, e) = Z. AF (( e H H ) o ) = AF ( H H e ) Id Fe. It may come as a surrise to the reader to find out that confusion regarding this issue was the original motivation for this aer. References [AGM] J. F. Adams, J. H. Gunawardena, and H. Miller, The Segal conjecture for elementary abelian - grous, Toology 24 (1985), no. 4, MR [AKO] M. Aschbacher, R. Kessar, and B. Oliver, Fusion systems in algebra and toology, London Mathematical Society Lecture Note Series, vol. 391, Cambridge University Press, Cambridge, MR (2012m:20015) [Bou1] A. K. Bousfield, The localization of saces with resect to homology, Toology 14 (1975), MR [Bou2], The localization of sectra with resect to homology, Toology 18 (1979), no. 4, [BK] A. K. Bousfield and D. M. Kan, Homotoy limits, comletions and localizations, Lecture Notes in Mathematics, Vol. 304, Sringer-Verlag, Berlin-New York, MR [BLO2] C. Broto, R. Levi, and B. Oliver, The homotoy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, MR (2004k:55016) [Car] G. Carlsson, Equivariant stable homotoy and Segal s Burnside ring conjecture, Ann. of Math. (2) 120 (1984), no. 2, MR (86f:57036) [CE] H. Cartan and S. Eilenberg, Homological algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, With an aendix by David A. Buchsbaum; Rerint of the 1956 original. MR [Che] A. Chermak, Fusion systems and localities, Acta Math. 211 (2013), no. 1, MR [DK] J. F. Davis and P. Kirk, Lecture notes in algebraic toology, Graduate Studies in Mathematics, vol. 35, American Mathematical Society, Providence, RI, MR [LMM] L. G. Lewis, J. P. May, and J. E. McClure, Classifying G-saces and the Segal conjecture, Current trends in algebraic toology, Part 2 (London, Ont., 1981), CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, MR (84d:55007a) [May] J. P. May, Stable mas between classifying saces, Conference on algebraic toology in honor of Peter Hilton (Saint John s, Nfld., 1983), Contem. Math., vol. 37, Amer. Math. Soc., Providence, RI, 1985, MR [MM] J. P. May and J. E. McClure, A reduction of the Segal conjecture, Current trends in algebraic toology, Part 2 (London, Ont., 1981), CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, MR (84d:55007b) [Rag] K. Ragnarsson, Classifying sectra of saturated fusion systems, Algebr. Geom. Tool. 6 (2006), MR (2007f:55013)

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