Saturated fusion systems as stable retracts of groups

Transcription

1 Massachusetts Institute of Technology Department of Mathematics Saturated fusion systems as stable retracts of groups Sune Precht Reeh MIT Topology Seminar, September 28, 2015 Slide 1/24

2 Outline 1 Bisets as stable maps 2 Fusion systems and idempotents 3 An application to HKR character theory (with T. Schlank & N. Stapleton) Notes on the blackboard are in red. Slide 2/24

3 Bisets Let S, T be finite p-groups. An (S, T )-biset is a finite set equipped with a left action of S and a free right action of T, such that the actions commute. Transitive bisets: [Q, ψ] T S := S T/(sq, t) (s, ψ(q)t) for Q S and ψ : Q T. Q and ψ are determined up to preconjugation in S and postconjugation in T. (S, T )-bisets form an abelian monoid with disjoint union. The group completion is the Burnside biset module A(S, T ), consisting of virtual bisets, i.e. formal differences of bisets. The [Q, ψ] form a Z-basis for A(S, T ). Example for D 8 with subgroup diagram. With V 1 as one of the Klein four groups, Q 1 as a reflection contained in V 1, and Slide 3/24

4 Bisets 2 Z as the centre/half-rotation of D 8, we for example have [V 1, id] 2[Q 1, Q 1 Z] as an element of A(D 8, D 8 ). We can compose bisets : A(R, S) A(S, T ) A(R, T ) given by X Y := X S Y when X, Y are actual bisets. A(S, S) is the double Burnside ring of S. A special case of the composition formula: [Q, ψ] T S [T, ϕ]r T = [Q, ϕψ]r S. We can think of (S, T )-bisets as being a sort of morphism from S to T. Slide 4/24

5 Bisets as stable maps Theorem (Segal conjecture. Carlsson,... ) For p-groups S, T : [Σ + BS, Σ + BT ] A(S, T ) p = {X A(S, T ) p X / T Z}. Corollary (?) For p-groups S, T : [(Σ + BS) p, (Σ + BT ) p ] = A(S, T ) p. Slide 5/24

6 Fusion systems A fusion system over a finite p-group S is a category F where the objects are the subgroups P S and the morphisms satisfy: Hom S (P, Q) F(P, Q) Inj(P, Q) for all P, Q S. Every ϕ F(P, Q) factors in F as an isomorphism P ϕp followed by an inclusion ϕp Q. A saturated fusion system satisfies a few additional axioms that play the role of Sylow s theorems. The canonical example of a saturated fusion system is F S (G) defined for S Syl p (G) with morphisms for P, Q S. Hom FS (G)(P, Q) := Hom G (P, Q). Slide 6/24

7 Fusion systems 2 Example for D 8 Σ 4 : If V 1 consists of the double transpositions in Σ 4, then the fusion system F = F D8 (Σ 4 ) gains an automorphism α of V 1 of order 3, and Q 1 V 1 becomes conjugate in F to Z V 1. Slide 7/24

8 Characteristic bisets If G induces a fusion system on S, we can ask what properties G has as an (S, S)-biset in relation to F S (G). Linckelmann-Webb wrote down the essential properties as the following definition: An element Ω A(S, S) p is said to be F-characteristic if Ω is left F-stable: res ϕ Ω = res P Ω in A(P, S) p for all P S and ϕ F(P, S). Ω is right F-stable. Ω is a linear combination of transitive bisets [Q, ψ] S S with ψ F(Q, S). Ω / S is invertible in Z (p). Slide 8/24

9 Characteristic bisets 2 G, as an (S, S)-biset, is F S (G)-characteristic. Σ 4 as a (D 8, D 8 )-biset is isomorphic to Σ 4 = [D8, id] + [V 1, α]. This biset is F D8 (Σ 4 )-characteristic. On the other hand, the previous example [V 1, id] 2[Q 1, Q 1 Z] is generated by elements [Q, ψ] with ψ F, but it is not F-stable and hence not characteristic. We prefer a characteristic element that is idempotent in A(S, S) p. Slide 9/24

10 Characteristic bisets 3 Theorem (Ragnarsson-Stancu) Every saturated fusion system F over S has a unique F-characteristic idempotent ω F A(S, S) (p) A(S, S) p, and ω F determines F. For the fusion system F = F D8 (Σ 4 ), the characteristic idempotent takes the form ω F = [D 8, id] [V 1, α] 1 3 [V 1, id]. Slide 10/24

11 BF as a stable retract of BS The characteristic idempotent ω F A(S, S) p for a saturated fusion system F defines an idempotent selfmap Σ + BS ω F Σ + BS. This splits off a direct summand W of Σ + BS, with properties: If F = F S (G), then W Σ + (BG p ). Each F has a classifying space BF, and W Σ + BF. We have BF BG p when F = F S (G). Have maps i: Σ + BS Σ + BF and tr: Σ + BF Σ + BS s.t. i tr = id Σ + BF and tr i = ω F. Slide 11/24

12 BF as a stable retract of BS Each saturated fusion system F over a p-group S corresponds to the retract Σ + BF of Σ + BS. Strategy Consider known results for finite p-groups. Apply ω F everywhere. Get theorems for saturated fusion systems, and p-completed classifying spaces. Slide 12/24

13 HKR character theory Hopkins-Kuhn-Ravenel constructed a generalization of group characters in Morava E-theory: χ n : E n(bg) Cl n,p (G; L(E n)). L(E n) is a certain algebra over E n. Cl n,p (G; L(E n)) contains functions valued in L(E n) defined on G-conjugacy classes of n-tuples of commuting elements in G of p-power order. Theorem (Hopkins-Kuhn-Ravenel) L(E n) E n E n(bg) Cl n,p (G; L(E n)). Slide 13/24

14 A further generalization by Stapleton gives character maps En(BG) C t LK(t) En 0 L K(t)En(Λ n t p BG). C t is of chromatic height t and an algebra over L K(t) E 0 n (and E 0 n). The r-fold free loop space Λ r BG decomposes as a disjoint union of centralizers: Λ r BG C G (α). α commuting r-tuple in G up to G-conj Λ r pbg is the collection of components for commuting r-tuples of elements of p-power order. Slide 14/24

15 Theorem (Stapleton) C t E 0 n En(BG) C t LK(t) En 0 L K(t)En(Λ n t p BG). The case t = 0 is the HKR character map. Slide 15/24

16 HKR character theory for fusion systems joint with Tomer Schlank & Nat Stapleton We consider the character map for p-groups E n(bs) C t LK(t) E 0 n L K(t)E n(λ n t BS) and try to make ω F act on both sides in a way that commutes with the character map. If successful, we get: Conjecture/Pretheorem (R.-Schlank-Stapleton) For every saturated fusion system F we have C t E 0 n E n(bf) C t LK(t) E 0 n L K(t)E n(λ n t BF). For F = F S (G) this recovers the theorem for finite groups. Slide 16/24

17 The proof Let Λ := Z/p k for k 0. Think of Λ as emulating S 1. The character map can be decomposed as En(BS) ev En(BΛ n t Λ n t BS) E n(bλ n t ) E n E n(λ n t BS) C t E 0 n E n(λ n t BS) C t LK(t) E 0 n L K(t)E n(λ n t BS) The first map is induced by the evaluation map ev : BΛ n t Λ n t BS BS. Slide 17/24

18 The proof 2 With the decomposition Λ r BS Commuting r-tuples a in S up to S-conjugation BC S (a), the evaluation map can be described algebraically as (Z/p k ) r C S (a) S given by (t 1,..., t r, s) (a 1 ) t1 (a r ) tr z. Slide 18/24

19 Consider functoriality of the evalutation map ev BΛ r Λ r ev BS BS? f BΛ r Λ r BT ev BT If f is a map BS BT of spaces, then we can just plug in id Λ r (f) into the square. However, if f is a stable map, such as ω F, we can t apply Λ r ( ) to f. Slide 19/24

20 Pretheorem (R.-S.-S.) There is a functor M defined on suspension spectra of p-groups and saturated fusion systems, such that for each stable map f : Σ BS Σ BT the following square commutes: BΛ r Λ r ev BS BS : Stable maps M(f) BΛ r Λ r BT ev f BT Note: M(f) maps between coproducts of p-groups and fusion systems, so M(f) is a matrix of virtual bisets. Slide 20/24

21 For most stable maps f, it is impossible for M(f) to have the form id (Z/p k ) r (?). Hence the cyclic factor needs to be used nontrivially. The free loop space Λ r BF for a saturated fusion system, also decomposed as a disjoint union of centralizers: Proposition (Broto-Levi-Oliver) Λ r BF Commuting r-tuples a in S up to F-conjugation BC F (a) If AF p is the category of formal coproducts of p-groups and fusion systems, where maps a matrices of virtual bisets, then M is a functor from AF p to itself. Slide 21/24

22 E n(bt ) ev E n(bλ n t Λ n t BT ) f M(f) E n(bs) ev E n(bλ n t Λ n t BS) E n(bλ n t ) E n E n(λ n t BT ) not E n(bλ n t ) E n E n(λ n t BS) C t E 0 n E n(λ n t BT ) exists but mysterious id? C t E 0 n E n(λ n t BS) C t LK(t) E 0 n L K(t)E n(λ n t BT ) id conjectured C t LK(t) E 0 n L K(t)E n(λ n t BS) In fact we just need the case f = trt S for T S finite abelian p-groups. Slide 22/24

23 Thank you! Slide 23/24

24 References [1] Kári Ragnarsson and Radu Stancu, Saturated fusion systems as idempotents in the double Burnside ring, Geom. Topol. 17 (2013), no. 2, MR [2] Sune Precht Reeh, Transfer and characteristic idempotents for saturated fusion systems, 41 pp., preprint, available at arxiv: [3] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), no. 3, MR (2001k:55015) [4] Nathaniel Stapleton, Transchromatic generalized character maps, Algebr. Geom. Topol. 13 (2013), no. 1, MR Slide 24/24