ABSTRACT
Floer homology is a version of Morse theory for certain infinite-dimensional spaces. Morse theory and Floer homology are examples of topological field theories in dimensions one and two, respectively. This chapter focuses on certain infinite-dimensional manifolds which carry natural Morse functions which have infinite Morse indices. It looks at some examples heuristically using finite-dimensional approximations. The “homology groups” which are obtained in this way do not describe the topology of the underlying space in the ordinary sense. The chapter discusses Morse theory from a different point of view, namely, as an example of a general strategy for producing topological invariants of manifolds. It also discusses Floer’s construction of homology groups, which is very similar to the approach to Morse theory. The chapter considers Floer homology for a rather different functional, namely, the Chern-Simons functional on gauge equivalence classes of connections on a three-manifold.
