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

Let F:={fx: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.