A K-theoretical invariant and bifurcation for a parameterized family of functionals

Let F : = { f x : x ∈ X } be a family of functionals defined on a Hilbert manifold E ˜ and smoothly parameterized by a compact connected orientable n-dimensional manifold X, and let σ : X → E ˜ be a smooth section of critical points of F . The aim of this paper is to give a sufficient topological condition on the parameter space X which detects bifurcation of critical points for F from the trivial branch. Finally we are able to give some quantitative properties of the bifurcation set for perturbed geodesics on semi-Riemannian manifolds.