Spin-structures and 2-fold coverings

Abstract

We prove that the existence of a Spin-structure on an oriented realvector bundle and the number of them can be obtained in terms of 2-fold coverings of the total space of the SO(n)-principal bundle associated to the vector bundle. Basically we use theory of covering spaces. We give a few elementary applications making clear that the Spin-bundle associated to a Spin-structure is not suficient to classify such structure, as pointed out by [6].

Keywords

• Buldles
• principal bundles
• orientable bundles
• spin-structure
• fundamental group
• Stiefel-Whithney classes.