# The FKMM-invariant in low dimension

## TL;DR

This paper introduces the FKMM-invariant, a cohomological characteristic class that fully classifies 'Quaternionic' vector bundles in low dimensions, specifically for dimensions up to 3.

## Contribution

It defines and studies the FKMM-invariant, providing a classification tool for 'Quaternionic' vector bundles in low-dimensional topology.

## Key findings

- FKMM-invariant takes values in relative equivariant Borel cohomology
- The invariant completely classifies 'Quaternionic' vector bundles in dimensions ≤ 3
- Discussion on the surjectivity of the FKMM-invariant

## Abstract

In this paper we investigate the problem of the cohomological classification of "Quaternionic" vector bundles in low-dimension ($d\leqslant 3$). We show that there exists a characteristic classes $\kappa$, called the FKMM-invariant, which takes value in the relative equivariant Borel cohomology and completely classifies "Quaternionic" vector bundles in low-dimension. The main subject of the paper concerns a discussion about the surjectivity of $\kappa$.