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

• f 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

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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.

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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(BG) → Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−t p

BG). Ct is of chromatic height t and an algebra over LK(t)E0

n (and E0 n).

The r-fold free loop space ΛrBG decomposes as a disjoint union of centralizers: ΛrBG ≃

• α commuting r-tuple

in G up to G-conj

CG(α). Λr

pBG is the collection of components for commuting r-tuples of elements

• f p-power order.

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Theorem (Hopkins-Kuhn-Ravenel, Stapleton)

Ct ⊗E0

n E∗

n(BG) ≃

− → Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−t p

BG). The case t = 0 is the original HKR character map.

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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(BF) → Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−tBF),

so that tensoring with Ct gives an isomorphism Ct ⊗E0

n E∗

n(BF) ≃

− → Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−tBF).

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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:

• HomS(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) ∈ Sylp(AutF(S))). The canonical example of a saturated fusion system is FS(G) defined for S ∈ Sylp(G) with morphisms HomFS(G)(P, Q) := HomG(P, Q) for P, Q ≤ S. Example for D8 ≤ Σ4: If V1 consists of the double transpositions in Σ4, then the fusion system F = FD8(Σ4) gains an automorphism α of V1 of order 3, and Q1 ≤ V1 becomes conjugate in F to Z ≤ V1.

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Each saturated fusion system has an associated classifying space BF, which is not the geometric realization |F| ≃ ∗. This is due to Broto-Levi-Oliver, Chermak, Glauberman-Lynd. For a fusion system FS(G) realized by a group, we have BF ≃ BG∧

p .

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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 D8 with subgroup diagram. With V1 as one of the Klein four groups, Q1 as a reflection contained in V1, and Z as the centre/half-rotation

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• f D8, we for example have [V1, id] − 2[Q1, Q1 → Z] as an element of

A(D8, D8). 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.

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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: [Σ∞

+ BS, Σ∞ + BT] ≈ A(S, T)∧ p ∼

= {X ∈ A(S, T)∧

p | |X|/|T| ∈ Z}.

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If G induces a fusion system on S, we can ask what properties G has as an (S, S)-biset in relation to FS(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: resϕ Ω = resP 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.

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G, as an (S, S)-biset, is FS(G)-characteristic. Σ4 as a (D8, D8)-biset is isomorphic to Σ4 ∼ = [D8, id] + [V1, α]. This biset is FD8(Σ4)-characteristic. On the other hand, the previous example [V1, id] − 2[Q1, Q1 → Z] is generated by elements [Q, ψ] with ψ ∈ F, but it is not F-stable and hence not characteristic.

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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 ωF ∈ A(S, S)(p) ⊆ A(S, S)∧

p , and F can be recovered from ωF.

For the fusion system F = FD8(Σ4), the characteristic idempotent takes the form ωF = [D8, id] + 1

3[V1, α] − 1 3[V1, id].

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The characteristic idempotent ωF ∈ A(S, S)∧

p for a saturated fusion

system F defines an idempotent selfmap Σ∞

+ BS ωF

− − → Σ∞

+ BS.

This splits off Σ∞

+ BF as a direct summand of Σ∞ + BS.

We have maps i: Σ∞

+ BS → Σ∞ + BF and tr: Σ∞ + BF → Σ∞ + BS s.t.

i ◦ tr = idΣ∞

+ BF and tr ◦i = ωF.

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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 ωF everywhere.
• Get theorems for saturated fusion systems, and p-completed

classifying spaces.

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We consider the HKR character map for p-groups E∗

n(BS) → Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−tBS).

By making ωF act on both sides in a way that commutes with the character map, we get a character map for BF and an isomorphism

Theorem (R.-Schlank-Stapleton)

For every saturated fusion system F we have Ct ⊗E0

n E∗

n(BF) ≃

− → Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−tBF).

For F = FS(G) this recovers the theorem for finite groups.

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We go further and show that

Theorem (R-S-S)

E∗

n(BS) → Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−tBS) is a natural in BS for all

virtual bisets in A(T, S)∧

p and for all p-groups.

Let Λ := Z/pk for k ≫ 0. Think of Λ as emulating S1. The character map can be decomposed as E∗

n(BS) ev∗

− − → E∗

n(BΛn−t × Λn−tBS) ≃ E∗ n(BΛn−t) ⊗E∗

n E∗

n(Λn−tBS)

→ Ct ⊗E0

n E∗

n(Λn−tBS) → Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−tBS)

The first map is induced by the evaluation map ev: BΛn−t × Λn−tBS → BS.

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With the decomposition ΛrBS ≃

• Commuting r-tuples a in S

up to S-conjugation

BCS(a), the evaluation map can be described algebraically as (Z/pk)r × CS(a) → S given by (t1, . . . , tr, s) → (a1)t1 · · · (ar)tr · z.

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Consider functoriality of the evalutation map ev BΛr × ΛrBS BS BΛr × ΛrBT BT ev ? f ev If f is a map BS → BT of spaces, then we can just plug in id × Λr(f) into the square. However, if f is a stable map, such as ωF, we can’t directly apply Λr(−) to f. In the case of a transfer map from S to a subgroup T, there is a reasonable definition of transfer Λr(trS

T ) from ΛrBS to ΛrBT, which emulates the

induction formula for characters of representations. This transfer conjugates

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r-tuples in S to r-tuples in T ≤ S whenever possible. However idΛr × Λr(trS

T ) doesn’t commute with the evaluation maps.

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Theorem (R-S-S)

There is a functor M defined for suspension spectra of p-groups and saturated fusion systems, such that for each stable map f : Σ∞BS → Σ∞BT (up to homotopy) the following square commutes: BΛr × ΛrBS BS BΛr × ΛrBT BT ev M(f) f ev : Stable maps Note: M(f) maps between coproducts of p-groups and fusion systems, so M(f) is a matrix of virtual bisets.

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For most stable maps f, it is impossible for M(f) to have the form id(Z/pk)r × (?). Hence the cyclic factor needs to be used nontrivially. The free loop space ΛrBF for a saturated fusion system, also decomposes as a disjoint union of centralizers:

Proposition (Broto-Levi-Oliver)

ΛrBF ≃

• Commuting r-tuples a in S

up to F-conjugation

BCF(a) If AFp 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 AFp to itself.

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E∗

n(BT)

E∗

n(BS)

E∗

n(BΛn−t × Λn−tBT)

E∗

n(BΛn−t × Λn−tBS)

E∗

n(BΛn−t) ⊗E∗

n E∗

n(Λn−tBT)

E∗

n(BΛn−t) ⊗E∗

n E∗

n(Λn−tBS)

Ct ⊗E0

n E∗

n(Λn−tBT)

Ct ⊗E0

n E∗

n(Λn−tBS)

Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−tBT)

Ct ⊗LK(t)E0

n LK(t)E∗

n(Λn−tBS)

ev∗ (trS

T )∗

ev∗ M(trS

T )∗ not − ⊗ −

Λn−t(trS

T )∗

≃ ≃

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References

[1] K´ ari Ragnarsson and Radu Stancu, Saturated fusion systems as idempotents in the double Burnside ring, Geom. Topol. 17 (2013), no. 2, 839–904. MR3070516 [2] Sune Precht Reeh, Transfer and characteristic idempotents for saturated fusion systems, Adv. Math. 289 (2016), 161–211. MR3439684 [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, 553–594. MR1758754 (2001k:55015) [4] Nathaniel Stapleton, Transchromatic generalized character maps, Algebr. Geom.

• Topol. 13 (2013), no. 1, 171–203. MR3031640

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