Clifford module invariants of spin bundles

Abstract

In this paper, we study $KO$-theory invariants of Spin bundles obtained by the $\alpha$-construction from Clifford module representations of the Spinor group. We begin by describing their elementary properties including various Whitney sum formulae and their relation with the $d$-invariant for vector bundles over spheres. We next observe an important difference between the two half-Spin representations and then proceed to investigate the fiber homotopy properties of the invariants. We conclude with some applications.