Floer Homotopy of Lagrangian Submanifolds

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1 Floer Homotopy of Lagrangian Submanifolds (revisiting Cohen-Jones-Segal after 25 years) Mohammed Abouzaid Partly describing joint work with - A. Blumberg and T. Kragh - A. Blumberg. July 18, 2017

2 Symplectic topology context (X, ω) a closed symplectic manifold, with an embedded Lagrangian L. Fix a basepoint on L. The classifying maps of tangent spaces L BO and X BU induce a diagram Ω 2 X Ω 2 (X, L) ΩL ΩX Ω(X, L) Z BU Z BO O U U/O where I have used Bott periodicity in the second row, Ω(X, L) is the space of paths in X, starting at the basepoint, and ending on L, and Ω 2 (X, L) is its based loop space. In particular the map Ω 2 (X, L) Z BO is a map of A spaces. The corresponding Thom space is an A ring spectrum. Ω 2 (X, L) ν

3 Positive energy monoid We shall think of Ω 2 (X, L) ν as graded by π 2 (X, L), which has a map ω : π 2 (X, L) R. Denote by π 2 + (X, L) the monoid of non-negative area elements. There is a corresponding ring spectrum Ω 2 (X, L) ν +. Fundamental analogy Commutative algebra Symplectic topology Field of fractions π 2 (X, L) Ring with valuation π 2 + (X, L) Maximal ideal Classes of strictly positive area Residue field Classes of trivial area

4 Formulation of Floer homotopy Conjecture Assume that ω : π 2 (X ) R vanishes (symplectically aspherical). 1 The moduli spaces of holomorphic discs determine a curved A deformation of Ω 2 (X, L) ν +. 2 The corresponding category of operadic modules is an invariant of the pair (X, L) up to symplectomorphism (a map from Ω Symp(X ) to automorphisms). 3 If L and L are Hamiltonian isotopic, the bimodule Ω 2 (X, L, L ) ν deforms to a bimodule inducing an equivalence of module categories after localisation at the maximal ideal (a map from Ω Ham(X ) to natural transformations of the identity). Each of these statements has an analogue in Fukaya, Oh, Ohta, and Ono s theory of Lagrangian Floer cohomology.

5 The general case If X is not symplectically aspherical, we need to use the fact that Ω 2 (X, L) lifts to an A -algebra in the category of modules over the E 2 algebra Ω 2 (X ). The basic idea is then to use this E 2 algebra as the coefficients with respect to which Floer theory is defined: Conjecture 1 The moduli spaces of holomorphic spheres (with framing) determine a curved framed E 2 deformation of Ω 2 (X ) ν +. 2 The moduli spaces of holomorphic discs define a deformation of Ω 2 (X, L) ν + as an A algebra over Ω 2 (X ) ν +. 3 Repeat the story from the aspherical case.

6 Koszul duality An important question is when the (stable) homotopy type of a Lagrangian admits a deformation. The starting point is Koszul duality: the projection L pt makes the sphere spectrum S into a module over the suspension spectrum of ΩL (c.f. Cohen, Ganatra, Lekili, and Royer s talks). Theorem (Rothenberg-Steenrod, Dwyer-Greenlees-Iyengar) There is a canonical equivalence of ring spectra L TL = End Σ ΩL(S). The Thom space of the normal bundle of L is the analogue of the cochains, and thus should be the right object to appear in a spectral version of the Fukaya category. The key condition is therefore to make sure that the twists are trivial, i.e. that the Thom spectra discussed earlier are equivalent to (shifted) suspension spectra of the underlying spaces.

7 Trivial spherical fibrations X is symplectically aspherical. The condition that Ω 2 (X, L) ν be equivalent to the suspension spectrum is the existence of a null homotopy of the classifying map of the stable spherical fibration: Ω 2 (X, L) Z BO BGL 1 (S). We need a trivialisation compatible with products, i.e. a null homotopy (c.f. Jin and Treumann) Ω(X, L) B 2 GL 1 (S). In this case there are compatible trivialisation of the tangent bundles of the Floer-Gromov moduli space of discs in L, as stable spherical fibrations, relative the pullback of TL under evaluation. Remark Smooth manifolds are unnecessary for the construction of Floer homotopy theory, and even topological manifolds carry more information than strictly needed.

8 The p-homotopy type Conjecture The ring spectrum L TL admits an (ungraded) curved A deformation whenever the above map is null-homotopic. The homotopy groups of GL 1 (S) are finite in each degree. Since L and X are finite complexes, this implies that the map Ω(X, L) B 2 GL 1 (S) admits a null homotopy after inverting finitely many primes. Thus, for each Lagrangian in an aspherical symplectic manifold, the orientation obstructions to constructing a multiplicative (stable) Floer p-homotopy type vanish except for finitely many p.

9 These statements are about stable homotopy types, without any multiplicative structure. The chromatic homotopy type Dropping the assumption that X is symplectically aspherical, we expect Floer-Gromov moduli spaces to be (virtual) orbifolds. Let p be a prime, n a strictly positive integer, and let K(n) be the height-n Morava theory. Work of Ravenel implies that K(n)-oriented orbifolds satisfy Poincaré duality with respect to K(n) (Greenlees-Sadofsky, Kuhn, Cheng,...): Conjecture If the map Ω(X, L) B 2 Aut S (K(n)) is null-homotopic (if p 3 and L is Pin?), L has a (curved) Floer K(n)-homotopy type. If L is the diagonal, the above map is null-homotopic for all p?. Conjecture The K(n)-homotopy type of a symplectic manifold is well-defined.

10 The A operad Everybody in this room has their favourite model of an A operad O, with spaces O(n) given by: Moduli spaces of stable discs with n + 1 boundary marked points. Cellular complexes with cells index by rooted planar trees with n leaves which are stable (i.e. vertices have valence 3). Any cofibrant resolution of the associative operad. We obtain an A operad in spectra by suspension. More familiar in Floer theory is the corresponding operad in chain complexes. Using cellular chains of the Stasheff associahedra, we get a generator µ n of degree n 2 in O(n) for 2 n, whose image under the differential is the sum of products µ k j µ n k+1 over the operadic structure maps. Remark In this story, the differential is not part of the operadic structure maps.

11 Curved A algebras A curved A algebra a chain complex together with a collection of operations {µ n } n 0 of degree n 2 satisfying a natural quadratic relation. The zeroth operation is an element µ 0 of degree 2. Definition The naive curved A operad (of k modules) is constructed from (rooted planar) trees with vertices of arbitrary valency (i.e. including 2 and 1). Lemma The naive curved A operad is acyclic. Corollary (Folklore going back to Kontsevich?) The homotopy theory of curved A algebras is trivial.

12 Gapped curved A algebras We know from the work of Fukaya, Oh, Ohta, and Ono that there are meaningful examples of curved A algebras, which they call gapped: work with filtered complexes (say, by positive integers), assume that we start with a usual A algebra, modulo the positive part of the filtration. The curvature then lives only in the higher parts. We can assign to curved A algebras a category of modules which is an honest A category (see Positselski). The work of Fukaya, Oh, Ohta, and Ono considers a special type of such modules obtained from bounding cochains. We want to do the same with spectra.

13 Curved A operad graded by monoids Definition A (f.g.) commutative monoid Γ is gapped if, for each γ Γ, there are only finitely many non-degenerate solutions to the equation γ i = γ. The fundamental example is the submonoid of classes in H 2 (X, L) which are represented by stable holomorphic discs. FALSE. ONLY TRUE UP TO BOUNDED ENERGY A tree with vertices labelled by Γ is stable if the univalent and bivalent vertices have non-zero labels. Such trees define an operad O Γ (in chain complexes, or k-modules for a ring spectrum), which we call the Γ-graded curved A operad (more complicated definition for non-commutative monoids such as π 2 (X, L)). Remark The operad O Γ does not come from an operad in spaces, because of the presence of negative dimensional cells (bivalent and univalent vertices), but rather from an operad in stacks as in Behrend-Manin

14 The based disc space as a curved algebra The moduli spaces of stable holomorphic discs with boundary on a Lagrangian in a symplectically aspherical manifold have a forgetful map to the curved A operad graded by the positive part of (a submonoid of) π 2 (M, L). Using an analogue of the Chas-Sullivan product (c.f. Royer), we expect to use these moduli spaces to make Ω 2 (X, L) ν + into a curved A algebra. Then, using the standard theory of (non-symmetric) operadic modules, one associates to a Lagrangian L the category of modules for the O Γ -algebra Ω 2 (X, L) ν +. This is the analogue of the Floer homology over the Novikov ring.

15 Localisation at the maximal ideal In Floer theory, the construction of a homology group with the right properties with respect to Hamiltonian isotopies requires inverting the Novikov parameter. We propose to implement this via a categorical quotient: let Ω 2 (X, L) ν 0 denote the spectrum associated to the kernel of the area map. The quotient by the maximal ideal gives a ring map Ω 2 (X, L) ν + Ω 2 (X, L) ν 0, and it is easy to see that this makes Ω 2 (X, L) ν 0 into a module with respect to the O Γ structure as well. Definition (see Positselski) The Floer homotopy category of the pair (X, L) is the quotient of the category of modules by the subcategory generated by Ω 2 (X, L) ν 0.