Isomorphism and stable isomorphism in "real" and "quaternionic" K-theory

TL;DR

This paper establishes lower bounds on the rank of 'real' and 'quaternionic' vector bundles over involutive spaces, impacting the classification of topological insulators with time-reversal symmetry.

Contribution

It provides new lower bounds for 'real' and 'quaternionic' bundles that influence their classification and the understanding of stable isomorphism in K-theory.

Findings

Lower bounds on vector bundle ranks established

Implications for topological insulator classification

Stable isomorphism conditions clarified

Abstract

We find lower bounds on the rank of a "real" vector bundle over an involutive space, such that "real" vector bundles of higher rank have a trivial summand and such that a stable isomorphism for such bundles implies ordinary isomorphism. We prove similar lower bounds also for "quaternionic" bundles. These estimates have consequences for the classification of topological insulators with time-reversal symmetry.