Real secondary index theory

Abstract

In this paper, we study the family index of a family of spin manifolds. In particular, we discuss to what extent the real index (of the Dirac operator of the real spinor bundle if the fiber dimension is divisible by $8$) which can be defined in this case contains extra information over the complex index (the index of its complexification). We study this question under the additional assumption that the complex index vanishes on the $k$–skeleton of $B$. In this case, we define new analytical invariants ${ĉ}_k\in H^{k-1}\left(B;ℝ∕\mathbb{Z}\right)$, certain secondary invariants.

We give interesting nontrivial examples. We then describe this invariant in terms of known topological characteristic classes.