On the existence of N ∞ operads in equivariant homotopy theory David White Denison University Joint with Javier Guti´ errez June 20, 2019 David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
Exotic Smooth Structures on Spheres Classical: If two smooth manifolds are homeomorphic, are they diffeomorphic? Answer (Milnor, 1956): No! S 7 has exotic smooth structures. Next question: Can we classify all exotic smooth structures on spheres, S n ? (First, assume n � 4). Milnor and Kervaire (1963): the group of smooth n -dim manifolds homeomorphic to S n (under connect sum operation) is isomorphic to the group Θ n of h -cobordism classes of homotopy n -spheres. Note: an h -cobordism is a cobordism M ֒ → W ← ֓ N where the inclusions are homotopy equivalences. Note: M is a homotopy sphere iff M is an h -cobordism sphere (Smale et. al) iff M is a topological sphere (by Perelman). David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
Framed Manifolds and bP n + 1 ≤ Θ n Θ n = h -cobordism classes of homotopy n -spheres; finite, abelian. Cyclic subgroup bP n + 1 ≤ Θ n of n -spheres that bound parallelizable manifolds. “Easy.” Parallelizable manifold has trivial tangent bundle (hence also trivial normal bundle). Framed means it has a chosen trivialization of the normal bundle. Kervaire-Milnor; Levine: 0 if n+1 is odd bP n + 1 = if n + 1 = 4 k C Bernoulli 0 or C 2 if n + 1 = 4 k + 2 bP n + 1 � C 2 = Z / 2 when n + 1 = 4 k + 2, 2 k + 1 � 2 ℓ − 1 David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
Kervaire Invariant and Framed Surgery bP n + 1 known except n + 1 = 2 j + 1 − 2 (where 0 or Z / 2), i.e. dim 2, 6, 14, 30, 62, 126, 254, . . . Recall J -homomorphism J : π n ( SO ( k )) → π n + k ( S k ) Framed surgery theory gives an injection (onto if n is odd) ψ Θ n / bP n + 1 ֒ → coker ( J ) = π s ∗ / im ( J ) . It’s an iso. iff Kervaire invariant in dim n is 0 (otherwise, image of index 2) Kervaire invariant (of n -dim framed manifold) is Arf invariant of the skew-symmetric pairing on the middle-dimensional homology. It’s an obstruction to framed surgery. David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
Pushing the problem into stable homotopy Browder (1969): K ( M n ) = 1 only possible if n = 2 j + 1 − 2. It’s 1 iff j ∈ Ext 2 , 2 j + 1 the class h 2 ( Z / 2 , Z / 2 ) persists to the E ∞ -page, i.e. A represents an element θ j ∈ π s 2 j + 1 − 2 There are M with K ( M ) = 1 in dim 2, 6, 14, 30, and 62. Exact sequence 0 → bP n + 1 → Θ n → π s ∗ / im ( J ) → 0 splits when n � 2 ℓ − 1 or 2 ℓ − 2. Φ Other n : 0 = bP n + 1 → Θ n → π s → C 2 → bP n → 0 ∗ / im ( J ) Extension problem: Φ iso. (if K ( M ) = 1) or bP n � 0. David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
Enter Hill, Hopkins, Ravenel (2009) Browder (1969 + computations in π s n ): Can only have Kervaire invariant 1 if n = 2 j + 1 − 2 = 2 , 6 , 14 , . . . j ∈ Ext 2 , 2 j + 1 HHR (2009): For j ≥ 7, the element h 2 ( Z / 2 , Z / 2 ) does A not represent an element θ j ∈ π s 2 j + 1 − 2 Corollary: Unless n = 2 , 6 , 14 , 30 , 62 , 126, there is no manifold of Kervaire invariant 1. So, only n = 126 is left! Corollary: In most dimensions, Θ n / bP n + 1 → π s ∗ / im ( J ) is an isomorphism. Corollary: Except in dimensions 2, 6, 14, 30, 62, and maybe 126, every stably framed smooth manifold is framed cobordant to a homotopy sphere. Surgery works! David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
Back to Exotic Smooth Structures For n � 4 , 125 , 126, if the order of π s n is known, we can compute the number of exotic n -spheres. Except for n of the form 2 k − 3 ≥ 125, we can also describe the group Θ n precisely. Example: For dimension n = 7, the group Θ 7 is the cyclic group Z Z / 28 Theorem (HHR): Unless n = 2 , 6 , 14 , 30 , 62 , 126, when n = 4 k + 2, Θ 4 k + 2 � π s 4 k + 2 , and when n = 4 k + 1, | Θ 4 k + 1 | = a k | π s 4 k + 1 | where a k = 1 if k even and 2 if k odd. Theorem (Wang-Xu): no exotic smooth structures in dim 5, 6, 12, 56, 61. Proof by computing π s n . David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
HHR proof sketch j ∈ Ext 2 , 2 j + 1 To show h 2 ( Z / 2 , Z / 2 ) does NOT represent A θ j ∈ π s 2 j + 1 − 2 : Create the 256-periodic spectrum (generalized cohomology 1 theory) Ω = D − 1 MU ∧ 4 . The Detection Theorem - can see if θ j is zero or not via its 2 Hurewicz image in Ω 2 − 2 j + 1 ( pt ) The Periodicity Theorem: Ω ∗ + 256 ( X ) � Ω ∗ ( X ) 3 The Gap Theorem: Ω i ( pt ) = 0 for − 4 < i < 0 4 Proof relies on Slice Spectral Sequence in G -spectra ( G = Z / 8). David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
Orthogonal G -spectra An orthogonal G -spectrum is a sequence ( X n ) of G × O ( n ) -spaces, with σ n : Σ X n → X n + 1 . Structure maps X n ∧ S k → X n + k are G × O ( n ) × O ( k ) -equivariant. Denote Sp G . Topological closed symmetric monoidal model category with Hom ( X , Y ) n = � m ≥ n Map O ( m − n ) ( X m − n , Y m ) . � ( X ∧ Y ) n = O ( n ) + ∧ O ( p ) × O ( q ) ( X p ∧ Y q ) p + q = n E is a commutative ring G -spectrum if τ : E ∧ E → E ∧ E , η : S → E , and associative, unital, commutative µ : E ∧ E → E (via commutative diagrams). Denote CAlg ( Sp G ) . David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
Multiplicative Norms for Commutative G -spectra Adjoint ind G H ( X ) ⊣ res G H : G -set → H -set; ind G H ( X ) = � G / H X For G -spectra, ind G H ( X ) = � i ∈ G / H ( H i ) + ∧ H X Can also define N G H ( X ) = ∧ i ∈ G / H ( H i ) + ∧ H X . For any finite G -set T , can define N T X = � T X . Adjunction ( N G H ⊣ res G H ) : CAlg ( Sp H ) ⇆ CAlg ( Sp G ) Commutative ring G -spectra X have multiplicative norm maps N T X → X for all T . These are used in the HHR computations that resolve the Kervaire Invariant One problem. Every homomorphism ρ : G → Σ | T | gives G ⋊ Σ action on N T X . Norm maps via G + ∧ H N T ( res H X ) � ( G × Σ n ) / Γ T ∧ Σ | T | X ∧| T | and X ∧| T | → X . David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
G -operads Operads encode algebraic structure. An operad P is a collection of sets (or spaces or G -spaces) P ( n ) parameterizing n -ary operations f : X ∧ n → X for all n . Action of Σ n on P ( n ) , unit 1 ∈ P ( 1 ) , and composition ◦ : P ( k ) × ( P ( n 1 ) × · · · × P ( n k )) → P ( n ) for n = � k i = 1 n i . Algebras X have P ( n ) ∧ Σ n X n → X for all n . Examples: Com has Com ( n ) = ∗ for all n . Algebras = CAlg ( Sp G ) . 1 In Top , E ∞ ( n ) = E Σ n (free Σ n -action and contractible), and 2 E ∞ -operads parameterize “homotopy coherent” commutativity. In Top G , N ∞ -operads encode E ∞ plus multiplicative norms. 3 David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
N ∞ -operads (Blumberg-Hill 2015) An N ∞ operad is a G -operad P such that P ( 0 ) is G -contractible, the action of Σ n on P ( n ) is free, and P ( n ) is the universal space for a family F n = N n ( P ) of subgroups of G × Σ n which contains all subgroups of the form H × 1. Here P ( n ) Γ = ∅ if Γ � F n , and P ( n ) Γ = ∗ otherwise. If F n is all subgroups of G × Σ n that contain all subgroups of the form H × 1, then you have all norms, and it’s complete N ∞ . These operads are G -weakly equivalent to Com . If F n = { H × 1 } , then N ∞ is the same as E ∞ in Top G . Motivating Question: Which collections F = ( F n ) have associated N ∞ -operads? David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
Model Categories A model category is a setting for abstract homotopy theory. Examples: Top, sSet, Ch(R), stable module cat, Spectra, G-spectra, motivic spectra, operads, categories, graphs, flows, ... Formally, a bicomplete category M and classes of maps W , F , Q (= weak equivalences, fibrations, cofibrations) satisfying axioms to behave like Top. Lifting, factorization, 2 out of 3, retracts. An object X is cofibrant if ∅ → X is a cofibration (where ∅ is initial). The cofibrant replacement QY of Y is the result of factoring ∅ → Y into cofibration followed by trivial fibration QY → Y . Ex: CW approximation, Projective Resolution. David White Denison University On the existence of N ∞ operads in equivariant homotopy theory
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