m i t m a t h e m a t i c s 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
m i t m a t h e m a t i c s 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
m i t m a t h e m a t i c s 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
m i t m a t h e m a t i c s 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
m i t m a t h e m a t i c s Bisets as stable maps Theorem (Segal conjecture. Carlsson, . . . ) For p -groups S, T : p ∼ [Σ ∞ + BS, Σ ∞ + BT ] ≈ A ( S, T ) ∧ = { X ∈ A ( S, T ) ∧ p | | X | / | T | ∈ Z } . Corollary (?) For p -groups S, T : p ] ∼ [(Σ ∞ + BS ) ∧ p , (Σ ∞ + BT ) ∧ = A ( S, T ) ∧ p . Slide 5/24
m i t m a t h e m a t i c s 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 Hom F S ( G ) ( P, Q ) := Hom G ( P, Q ) . for P, Q ≤ S . Slide 6/24
m i t m a t h e m a t i c s 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 D 8 (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
m i t m a t h e m a t i c s 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
m i t m a t h e m a t i c s 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 ∼ = [ D 8 , id ] + [ V 1 , α ] . This biset is F D 8 (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
m i t m a t h e m a t i c s 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 D 8 (Σ 4 ) , the characteristic idempotent takes the form ω F = [ D 8 , id ] + 1 3 [ V 1 , α ] − 1 3 [ V 1 , id ] . Slide 10/24
m i t m a t h e m a t i c s B F 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 ω F Σ ∞ → Σ ∞ + BS − − + 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” B F , and W ≃ Σ ∞ + B F . We have B F ≃ BG ∧ p when F = F S ( G ) . • Have maps i : Σ ∞ + BS → Σ ∞ + B F and tr: Σ ∞ + B F → Σ ∞ + BS s.t. i ◦ tr = id Σ ∞ + B F and tr ◦ i = ω F . Slide 11/24
m i t m a t h e m a t i c s B F as a stable retract of BS Each saturated fusion system F over a p -group S corresponds to the retract Σ ∞ + B F 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
m i t m a t h e m a t i c s 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) n ( BG ) ≃ L ( E ∗ n E ∗ → Cl n,p ( G ; L ( E ∗ n ) ⊗ E ∗ − n )) . Slide 13/24
m i t m a t h e m a t i c s A further generalization by Stapleton gives character maps n (Λ n − t E ∗ n L K ( t ) E ∗ n ( BG ) → C t ⊗ L K ( t ) E 0 BG ) . p 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 p BG is the collection of components for commuting r -tuples of elements of p -power order. Slide 14/24
m i t m a t h e m a t i c s Theorem (Stapleton) n ( BG ) ≃ n (Λ n − t n E ∗ n L K ( t ) E ∗ C t ⊗ E 0 − → C t ⊗ L K ( t ) E 0 BG ) . p The case t = 0 is the HKR character map. Slide 15/24
m i t m a t h e m a t i c s HKR character theory for fusion systems joint with Tomer Schlank & Nat Stapleton We consider the character map for p -groups n (Λ n − t BS ) E ∗ n L K ( t ) E ∗ n ( BS ) → C t ⊗ L K ( t ) E 0 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 n ( B F ) ≃ n (Λ n − t B F ) . n E ∗ n L K ( t ) E ∗ C t ⊗ E 0 − → C t ⊗ L K ( t ) E 0 For F = F S ( G ) this recovers the theorem for finite groups. Slide 16/24
m i t m a t h e m a t i c s The proof Let Λ := Z /p k for k ≫ 0. Think of Λ as emulating S 1 . The character map can be decomposed as n ( BS ) ev ∗ n ( B Λ n − t × Λ n − t BS ) E ∗ → E ∗ − − ≃ E ∗ n ( B Λ n − t ) ⊗ E ∗ n E ∗ n (Λ n − t BS ) n (Λ n − t BS ) n E ∗ → C t ⊗ E 0 n (Λ n − t BS ) n L K ( t ) E ∗ → C t ⊗ L K ( t ) E 0 The first map is induced by the evaluation map ev : B Λ n − t × Λ n − t BS → BS . Slide 17/24
m i t m a t h e m a t i c s The proof 2 With the decomposition Λ r BS ≃ � BC S ( a ) , Commuting r -tuples a in S up to S -conjugation 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 ) t 1 · · · ( a r ) t r · z. Slide 18/24
m i t m a t h e m a t i c s Consider functoriality of the evalutation map ev ev B Λ r × Λ r BS BS f ? ev B Λ r × Λ r BT 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
m i t m a t h e m a t i c s 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: ev B Λ r × Λ r BS BS M ( f ) f ev B Λ r × Λ r BT BT : Stable maps Note: M ( f ) maps between coproducts of p -groups and fusion systems, so M ( f ) is a matrix of virtual bisets. Slide 20/24
m i t m a t h e m a t i c s 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 B F for a saturated fusion system, also decomposed as a disjoint union of centralizers: Proposition (Broto-Levi-Oliver) Λ r B F ≃ � BC F ( a ) Commuting r -tuples a in S up to F -conjugation 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
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