Suppose we have a Fredholm section S, (the differential is Fredholm at the 0-set) of some Banach vector bundle over X, transverse to the 0-section, with Fredholm index 1 and such that the 0-set of this section is a circle which is contractible in X, can S be pushed of the 0-section through Fredholm sections?

edit: as stated this is not right. Mike Usher pointed out that in finite dimensional case (say a rank k real vector bundle over X^{k+1}) there are secondary obstructions to having a non-vanishing section. Since these obstructions have to do with $\pi_k (S^{k-1})$ it is not even obvious how to extend this to Fredholm setting. I guess the right question is then can one formulate obstruction theory in the Fredholm setting?

Hilbertrather thanBanach. The group of linear automorphisms of an arbitrary Banach space could be weird. Assumming this is the case, then your manifold $X$ is an open subset of a Hilbert space $H$, and The bundle is a trivializable Hilbert bundle. The section $S$ is then a map $S: X\to H$ and the section whose differential at every point is Fredholm of index $1$. – Liviu Nicolaescu May 9 '12 at 17:01