### Bifurcation of Fredholm maps I. The index bundle and bifurcation

#### Abstract

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.

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.

#### Keywords

Bifurcation; Fredholm maps; index bundle; J-homomorphism; elliptic BVP

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