A New Cohomology for the Morse Theory of Strongly Indefinite Functionals on Hilbert Spaces

Take a C function f :M → R on a complete Hilbert manifold which satisfies the Palais–Smale condition. Assume that it is a Morse function, meaning that the second order differential df(x) is non-degenerate at every critical point x. Recall that the Morse index m(x, f) of a critical point x is the dimension of the maximal subspace on which df(x) is negative definite. Then the basic result of Morse theory, as generalized by Palais [14], is the following: if c ∈ ]a, b[ is the only critical level in [a, b] and x0 is the only critical point at level c, then the set f b = {x ∈M | f(x) ≤ b} can be continuously deformed onto f ∪ Bm(x,f), where Bm(x,f) is an m(x, f)-dimensional closed ball, attached to f by its boundary. This local result is used to prove the well