A new light on the FKMM invariant and its consequences

TL;DR

This paper demonstrates that the FKMM invariant correctly classifies Quaternionic vector bundles in low dimensions by interpreting it through Bredon equivariant cohomology, advancing understanding of topological phases in quantum systems.

Contribution

It establishes the FKMM invariant as the fundamental characteristic class for Quaternionic vector bundles in dimensions up to three, using equivariant cohomology and homotopy theory.

Findings

FKMM invariant classifies Quaternionic bundles in low dimensions

Interpretation via Bredon equivariant cohomology provides new insights

Results align with expectations in topological quantum systems

Abstract

"Quaternionic" vector bundles are the objects which describe the topological phases of quantum systems subjected to an odd time-reversal symmetry (class AII). In this work we prove that the FKMM invariant provides the correct fundamental characteristic class for the classification of "Quaternionic" vector bundles in dimension less than, or equal to three (low dimension). The new insight is provided by the interpretation of the FKMM invariant from the viewpoint of the Bredon equivariant cohomology. This fact, along with basic results in equivariant homotopy theory, allows us to achieve the expected result.