  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.
