The floer homotopy type of the cotangent bundle

Abstract

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×T*M →R whichisquadraticnearinfinity. Z(T*M)isconstructed 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).