On Obstructions to K* ( ;Z/2) - Orientability PDF

Keywords

differentials, spherical fibration, cohomology

Abstract

The vanishing of the differentials of the Atiyah-Hirzebruch spectral sequence on the Thom class is the necessary and sufficient condition for O(n)- bundle to be orientable with respect to a cohomology theory h*. It is shown here that the third Stiefel-Whitney class is the first obstruction for an oriented spherical fibration to be orientable with respect to mod 2 K-theory K* ( ;Z/2). By killing the third differential of the Atiyah-Hirzebruch spectral sequence, we will find the second obstruction for an oriented real vector bundle to be orientable with respect to K* ( ;Z/2) by calculating the fifth differential in the Atiyah- Hirzebruch spectral sequence