Let M be a closed, oriented, n-dimensional manifold. In this paper we describe a spectrum in the sense of homotopy theory, Z(T^*M), whose homology is naturally isomorphic to the Floer homology of the cotangent bundle, T^*M. This Floer homology is taken with respect to a Hamiltonian H: S^1 x T^*M --> R, which is quadratic near infinity. Z(T^*M) is constructed assuming a basic smooth gluing result of J-holomorphic cylinders. This spectrum will have a C.W decomposition with one cell for every periodic solution of the equation defined by the Hamiltonian vector field X_H. Its induced cellular chain complex is exactly the Floer complex. The attaching maps in the C.W structure of Z(T^*M) are described in terms of the framed cobordism types of the moduli spaces of J -holomorphic cylinders in T^*M with given boundary conditions. This is done via a Pontrjagin-Thom construction, and an important ingredient in this is proving, modulo this gluing result, that these moduli spaces are compact, smooth, framed manifolds with corners. We then prove that Z(T^*M), which we refer to as the "Floer homotopy type" of T^*M, has the same homotopy type as the suspension spectrum of the free loop space, LM. This generalizes the theorem first proved by C. Viterbo that the Floer homology of T^*M is isomorphic to H_*(LM).
In 1995 the author, Jones, and Segal introduced the notion of "Floer homotopy theory". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, "When is the Floer homology isomorphic to the (singular) homology of a naturally occuring (pro)spectrum defined from the properties of the moduli spaces inherent in the Floer theory?". A proposal for how to construct such a spectrum was given in terms of a "framed flow category", and some rather simple examples were described. Years passed before this noti…
We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow (math/0604379, math/0612399) and the authors (math/0701783), before discussing a new approach using family Floer cohomology and the ``wrapped Fukaya category''. The latter, inspired by Viterbo's symplectic homology, emphasises the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply-connected case.
In this paper we describe and continue the study begun by the author, Jones, and Segal, of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a gen…
We construct a new cylinder object for semifree differential graded (dg) categories in the category of dg categories. Using this, we give a practical formula computing homotopy colimits of semifree dg categories. Combining it with the result of Ganatra, Pardon, and Shende, we get a formula computing wrapped Fukaya categories of Weinstein manifolds using their sectorial coverings. This formula has lots of applications including a practical computation of the wrapped Fukaya category of any cotangent bundle or plumbing space. In this paper, we compute wrapped Fukaya categories of cotangent bundle…
For a closed symplectic manifold $(M,ω)$, a compatible almost complex structure $J$, a 1-periodic time dependent symplectic vector field $Z$ and a homotopy class of closed curves $γ$ we define a Floer complex based on 1-periodic trajectories of $Z$ in the homotopy class $γ$. We suppose that the closed 1-form $i_{Z_t}ω$ represents a cohomology class $β(Z):=β$, independent of $t$. We show how to associate to $(M,ω,γ,β)$ and to two pairs $(Z_i,J_i)$, $i=1,2$ with $β(Z_i)=β$ an invariant, the relative symplectic torsion, which is an element in the Whitehead group $Wh(Λ_0)$, of a Novikov ring $Λ_0$…
We prove that the rank of the cohomology of a closed symplectic manifold with coefficients in a field of characteristic $p$ is smaller than the number of periodic orbits of any non-degenerate Hamiltonian flow. Following Floer, the proof relies on constructing a homology group associated to each such flow, and comparing it with the homology of the ambient symplectic manifold. The proof does not proceed by constructing a version of Floer's complex with characteristic $p$ coefficients, but uses instead the canonical (stable) complex orientations of moduli spaces of Floer trajectories to construct…
We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.
We construct "barcodes" for the chain complexes over Novikov rings that arise in Novikov's Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. Our barcodes completely characterize the filtered chain homotopy type of the chain complex; in particular they subsume in a natural way previous filtered Floer-theoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. We moreover prov…
Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz-Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalizing Khovanov cohomology. In this paper we give mo…
Inspired by Segal-Stolz-Teichner project for geometric construction of elliptic (tmf) cohomology, and ideas of Floer theory and of Hopkins-Lurie on extended TFT's, we geometrically construct some $Ring$-valued representable cofunctors on the homotopy category of topological spaces. Using a classical computation in Gromov-Witten theory due to Seidel we show that for one version of these cofunctors $π_{2}$ of the representing space is non trivial, provided a certain categorical extension of Kontsevich conjecture holds for the symplectic manifold $ \mathbb{CP} ^{n}$, for some some $n \geq 1$. Thi…
This is a research monograph on symplectic cohomology (disguised as an advanced graduate textbook), which provides a construction of this version of Hamiltonian Floer cohomology for cotangent bundles of closed manifolds. The focus is on the aspects of the theory that have been neglected in the literature: (1) the base is not assumed to be orientable or Spin, (2) local systems on the free loop space are used to define twisted versions of Floer cohomology, (3) a (twisted) Batalin-Vilkovisky structure is constructed, and (4) the BV relation is verified. In this setting (i.e. with all the appropri…
We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent index and handle cancellation which removes two critical points connected by a single flowline. A framed flow category is a way of encoding flow data such as that which may arise from the flowlines of a Morse function or of a Floer functional. The Cohen-Jones-Segal construction as…
Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We show that for a generic metric on such a smooth four manifold with second Betti number at least three, the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. The number of closed Reeb orbits of length at most l o…
We introduce a refinement of the Ozsvath-Szabo complex associated to a balanced sutured manifold $(X,τ)$ by Juhasz. An algebra $A_τ$ is associated to the boundary of a sutured manifold and a filtration of its generators by $H^2(X,\partial X;\Z)$ is defined. For a fixed Spin^c structure $s$ over the manifold $X'$, which is obtained from $X$ by filling out the sutures, the Ozsvath-Szabo chain complex $CF(X,τ,s)$ is then defined as a chain complex with coefficients in $A_τ$ and filtered by $\SpinC(X,τ)$. The filtered chain homotopy type of this chain complex is an invariant of $(X,τ)$ and the Spi…
We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams of simplicial categories. We present a classification of bundles of categories of spectra and of the associated twisted parametrized spectra. Though twisted parametrized spectra do not have global homotopy types and therefore do not have generalized homology invariants in the usua…
This is an expository paper about Seiberg-Witten Floer stable homotopy types. We outline their construction, which is based on the Conley index and finite dimensional approximation. We then describe several applications, including the disproof of the high-dimensional triangulation conjecture.
