Floer’s infinite dimensional Morse theory and homotopy theory

Abstract

\S 1 INTRODUCTION This paper is a progress reportl on our efforts to understand the ho-motopy theory underlying Floer homology. Its objectives are as follows: (A) to describe some of our ideas concerning what, exactly, the Floer homology groups compute; (B) to explain what kind of an object we think the 'Floer homotopy type' of an infinite dimensional manifold should be; (C) to work out, in detail, the Floer homotopy type in some examples. $\backslash Ve$ have not solved the problems posed by the underlying questions, but $\backslash ve$ do have aprogramme which we hope will lead to solutions. Thus it seems worthwhile to describe.our ideas now, especially in avolume of papers dedicated to the memory of Andreas Floer. $\backslash 4^{t}e$ plan to write acomplete account of this approach to Floer homotopy $t$ ,heory in afuture paper. Floer homology arises in two different contexts, the study of $c\iota lrves$ and surfaces in symplectic manifolds, and gauge theory on three-and four-dimensional manifolds. In each of these contexts there are two different perspectives, which one can think of as 'Hamiltonian' and 'Lagrangian'. The theory began with Floer's proof of the Arnold conjecture. On quite appropriate: Je n'ai fait celle-ci plus longue que parce que je n'ai pas eu le loisir de la faire plus courte. 883 4, 1994 G/68-96