When the Morse index is infinite

Abstract

Let f be a smooth Morse function on an infinite-dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and coindex. For any critical point x, choose an integer a(x arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the following property: for any pair of critical points x,y, the unstable manifold of x and the stable manifold of y have a transverse intersection of dimension a(x)−a(y.