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: 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).