Invariants of real vector bundles

TL;DR

This paper establishes a method to classify real vector bundles over a compact manifold with corners by using Dirac indices on a finite set of spin or spin^h manifolds mapped into it, providing a new invariant-based approach.

Contribution

It introduces a novel classification technique for real vector bundles using Dirac indices on selected spin or spin^h manifolds, enhancing understanding of bundle invariants.

Findings

Real vector bundles are classified up to stable equivalence by Dirac indices.

A finite set of spin or spin^h manifolds suffices for the classification.

The approach applies to manifolds with corners or finite CW-complexes.

Abstract

For a compact smooth manifold with corners (or finite CW-complex) $X$, we can prescribe a finite set of spin or spin$^h$ manifolds (possibly with boundary) mapping into it so that every real vector bundle over $X$ is determined, up to stable equivalence, by the Dirac indices of the real vector bundle when pulled-back onto those prescribed spin or spin$^h$ manifolds.