  Recently the first author studied the bifurcation of critical points of
families of functionals on a Hilbert space, which are parametrised by a compact
and orientable manifold having a non-vanishing first integral cohomology group.
We improve this result in two directions: topologically and analytically. From
the analytical point of view we generalise it to a broader class of
functionals; from the topological point of view we allow the parameter space to
be a metrisable Banach manifold. Our methods are in particular powerful if the
parameter space is simply connected. As an application of our results we
consider families of geodesics in (semi-) Riemannian manifolds.
