The index bundle for selfadjoint Fredholm operators and multiparameter bifurcation for Hamiltonian systems (Q6039904)

Summary: The index of a selfadjoint Fredholm operator is zero by the well-known fact that the kernel of a selfadjoint operator is perpendicular to its range. The Fredholm index was generalised to families by \textit{M. F. Atiyah} [K-theory. With notes by D. W. Anderson. Reprint. Redwood City, CA etc.: Addison-Wesley (1989; Zbl 0676.55006)] and \textit{K. Jänich} [in loc. cit.] in the sixties, and it is readily seen that, on complex Hilbert spaces, this so-called index bundle vanishes for families of selfadjoint Fredholm operators as in the case of a single operator. The first aim of this note is to point out that, for every real Hilbert space and every compact topological space \(X\), there is a family of selfadjoint Fredholm operators parametrised by \(X \times S^1\) which has a non-trivial index bundle. Further, we use this observation and a family index theorem of \textit{J. Pejsachowicz} [in: \(C^*\)-algebras and elliptic theory II. Selected papers of the international conference, Bȩdlewo, Poland, January 2006. Basel: Birkhäuser. 239--250 (2008; Zbl 1151.58008)] to study multiparameter bifurcation of homoclinic solutions of Hamiltonian systems, where we generalise a previously known class of examples.