Twisted spin structures

The purpose of this paper is to consider in greater detail one generalization of the notion of spin structure on an oriented vector bundle with zero w2 class to the case of all oriented vector bundles (with what ever w2); this generalization was introduced some what briefly in one of the author’s previous articles [1] . Detailed definitions for these generalized (or “twisted”) spin structures are given, and somegeneral properties are proved, analogous to the case of “regular” spin structures. In particular, a difference class for two “twisted” spin structures is constructed, which allows one to consider the set of all such structures as an “affine space” over the one dimensional mod 2 cohomology group of the base (quite similar again to the“regular” case).