Saturated fusion systems as stable retracts of groups (HKR - PowerPoint PPT Presentation

m i t m a t h e m a t i c s Saturated fusion systems as stable retracts of groups (HKR character theory for fusion systems) Sune Precht Reeh joint with Tomer Schlank & Nat Stapleton Alpine topology, Saas-Almagell, August 20, 2016 Slide 1/23

m i t m a t h e m a t i c s Outline 1 Motivation: The HKR character map 2 Background on fusion systems and bisets 3 Main theorem and the proof strategy 4 Transfer for free loop spaces Notes on the blackboard are in red . Sune Precht Reeh Slide 2/23

h k r c h a r a c t e r t h e o r y m i t m a t h e m a t i c s Fix a prime p . The HKR character map for Morava E-theory of a finite group was constructed by Hopkins-Kuhn-Ravenel, and generalized by Stapleton, as a map E ∗ n L K ( t ) E ∗ n (Λ n − t 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. Sune Precht Reeh Slide 2/23

h k r c h a r a c t e r t h e o r y m i t m a t h e m a t i c s Theorem (Hopkins-Kuhn-Ravenel, 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 original HKR character map. Sune Precht Reeh Slide 3/23

h k r c h a r a c t e r t h e o r y m i t m a t h e m a t i c s HKR character theory happens p -locally, so we might replace the finite group G with a saturated fusion system F at the prime p . We wish to define an HKR character map for F , E ∗ n L K ( t ) E ∗ n (Λ n − t B F ) , n ( B F ) → C t ⊗ L K ( t ) E 0 so that tensoring with C t gives an isomorphism n ( B F ) ≃ n E ∗ n L K ( t ) E ∗ n (Λ n − t B F ) . C t ⊗ E 0 − → C t ⊗ L K ( t ) E 0 Sune Precht Reeh Slide 4/23

f u s i o n s y s t e m s m i t m a t h e m a t i c s 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 (e.g. Inn( S ) ∈ Syl p (Aut F ( S ))). 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 . 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 . Sune Precht Reeh Slide 5/23

c l a s s i f y i n g s p a c e s f o r f u s i o n s y s t e m s m i t m a t h e m a t i c s Each saturated fusion system has an associated classifying space B F , which is not the geometric realization |F| ≃ ∗ . This is due to Broto-Levi-Oliver, Chermak, Glauberman-Lynd. For a fusion system F S ( G ) realized by a group, we have B F ≃ BG ∧ p . Sune Precht Reeh Slide 6/23

b i s e t s m i t m a t h e m a t i c s 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 Z as the centre/half-rotation Sune Precht Reeh Slide 7/23

b i s e t s m i t m a t h e m a t i c s 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 stable maps from BS to BT . [ Q, ψ ] T S is transfer from S to Q ≤ S followed by the map ϕ : Q → T . Sune Precht Reeh Slide 8/23

b i s e t s a s s t a b l e m a p s m i t m a t h e m a t i c s Virtual bisets give us all homotopy classes of stable maps between classifying spaces: Theorem (Segal conjecture. Carlsson, Lewis-May-McClure) For p -groups S, T : p ∼ [Σ ∞ + BS, Σ ∞ + BT ] ≈ A ( S, T ) ∧ = { X ∈ A ( S, T ) ∧ p | | X | / | T | ∈ Z } . Sune Precht Reeh Slide 9/23

c h a r a c t e r i s t i c b i s e t s m i t m a t h e m a t i c s 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 not divisible by p . Sune Precht Reeh Slide 10/23

c h a r a c t e r i s t i c b i s e t s m i t m a t h e m a t i c s 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. Sune Precht Reeh Slide 11/23

t h e c h a r a c t e r i s t i c i d e m p o t e n t m i t m a t h e m a t i c s We prefer a characteristic element that is idempotent in A ( S, S ) ∧ p . 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 F can be recovered from ω 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 ] . Sune Precht Reeh Slide 12/23

f u s i o n s y s t e m s a s s t a b l e r e t r a c t s o f p - g r o u p s m i t m a t h e m a t i c s 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 Σ ∞ + B F as a direct summand of Σ ∞ + BS . We 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 . Sune Precht Reeh Slide 13/23

f u s i o n s y s t e m s a s s t a b l e r e t r a c t s o f p - g r o u p s m i t m a t h e m a t i c s 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. Sune Precht Reeh Slide 14/23

h k r c h a r a c t e r t h e o r y f o r f u s i o n s y s t e m s m i t m a t h e m a t i c s We consider the HKR 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 By making ω F act on both sides in a way that commutes with the character map, we get a character map for B F and an isomorphism Theorem (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. Sune Precht Reeh Slide 15/23

t h e p r o o f m i t m a t h e m a t i c s We go further and show that Theorem (R-S-S) E ∗ n L K ( t ) E ∗ n (Λ n − t BS ) is a natural in BS for all n ( BS ) → C t ⊗ L K ( t ) E 0 virtual bisets in A ( T, S ) ∧ p and for all p -groups. 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 ∗ n ( B Λ n − t ) ⊗ E ∗ n (Λ n − t BS ) E ∗ → E ∗ n E ∗ − − n E ∗ n (Λ n − t BS ) → C t ⊗ L K ( t ) E 0 n L K ( t ) E ∗ n (Λ n − t BS ) → C t ⊗ E 0 The first map is induced by the evaluation map ev : B Λ n − t × Λ n − t BS → BS . Sune Precht Reeh Slide 16/23

t h e p r o o f m i t m a t h e m a t i c s 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. Sune Precht Reeh Slide 17/23