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

The author considers an orientable, compact, connected, n-dimensional manifold X and a family of functionals (fx)x∈X defined on a real separable Hilbert space H, fx:H→R such that the map (x,v)∈X×H↦fx(v)∈R is smooth, ∇fx(0)=0, for all x∈X, and the Fréchet second differential of fx at 0∈H is represented, with respect to the scalar product of H, by an indefinite bounded Fredholm operator Ax which depends smoothly on x. He gives a topological condition on X which implies the occurrence of bifurcation from the trivial branch of solutions for the equation ∇fx(v)=0, provided that there exists at least one x0∈X such that Ax0 is invertible. Such a topological condition is the nontriviality of a characteristic class, which the author identifies with the first Chern class of the analytical index of the family (Ax)x∈X. The analytical index, associated to the complexification of the family (Ax)x∈X, is a complex virtual vector bundle over the suspension of X; thus its first Chern class can be identified with an element of the cohomology group H1(X,Z). From the non-vanishing of this class the author deduces that the topological covering dimension of the bifurcation set is n−1. In the last section of the paper, bifurcation of perturbed geodesics on a semi-Riemannian manifold [cf. M. Musso, J. Pejsachowicz and A. Portaluri, ESAIM Control Optim. Calc. Var. 13 (2007), no. 3, 598--621; MR2329179 (2008e:58017)] is presented as an example of a geometrical problem that can be cast in the above setting.