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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? Note when the bundle is finite dimensional this is easily seen to be the case.

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?

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?

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? Note when the bundle is finite dimensional this is easily seen to be the case.

edited: thanks for the comment below, I wrote the question in a hurry, and tried to make it unnecessarily general. Above is exactly what I need.

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? Note when the bundle is finite dimensional this is easily seen to be the case.

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?

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 and such that the 0-set of this section is contractible in X, can S be pushed of the 0-section through Fredholm sections? Note when the bundle is finite dimensional this is easily seen to be the case.

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? Note when the bundle is finite dimensional this is easily seen to be the case.

edited: thanks for the comment below, I wrote the question in a hurry, and tried to make it unnecessarily general. Above is exactly what I need.