Bifurcation of Fredholm Maps I; The Index Bundle and Bifurcation

We associate to a parametrized family $f$ of nonlinear Fredholm maps possessing a trivial branch of zeroes an {\it index of bifurcation} $\beta (f)$ which provides an algebraic measure for the number of bifurcation points from the trivial branch. The index $\beta (f)$ is derived from the index bundle of the linearization of the family along the trivial branch by means of the generalized $J$-homomorphism. Using the Agranovich reduction and a cohomological form of the Atiyah-Singer family index theorem, due to Fedosov, we compute the bifurcation index of a multiparameter family of nonlinear elliptic boundary value problems from the principal symbol of the linearization along the trivial branch. In this way we obtain criteria for bifurcation of solutions of nonlinear elliptic equations which cannot be achieved using the classical Lyapunov-Schmidt method.