A Morse complex for infinite dimensional manifolds Part I

In this paper and in the forthcoming Part II we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M , possibly having critical points of infinite Morse index and co-index. The idea is to consider an infinite dimensional subbundle or more generally an essential subbundle of the tangent bundle of M , suitably related with the gradient flow of f . This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y: finite dimensionality of W , possibility that different components of W have different dimension, orientability of W and coherence in the choice of an orientation, compactness of the closure of W , classification, up to topological conjugacy, of the gradient flow on the closure of W , in the case dimW = 2.