2007/02/27 by Ralph L. Cohen, Cohen, Ralph L. · 1 citation
Mathematics · #53D40 #55P35 #55P42 #57R19 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.AT #math.SG #msc:53D40 #msc:55P35 #msc:55P42 #msc:57R19
paper · pdf · doi:10.48550/arxiv.math/0702852
36 pages. A gluing assumption is described, and a more complete discussion of framing issues is given
openalex publication_date 2007/02/27 · arxiv created 2007/08/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: S1 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 XH. 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).