Quaternionic structures

TL;DR

This paper explores modules over quaternion algebra bundles, discussing their properties and applications to geometric structures on manifolds, including obstructions to quaternionic structures.

Contribution

It provides a detailed account of modules over quaternion algebra bundles, including Morita equivalence, characteristic classes, and K-theory, with applications to geometry.

Findings

Obstructions for almost quaternionic structures on 8-dimensional Spinc manifolds identified

Modules over quaternion algebra bundles characterized and classified

Connections to quaternionic and algebraic geometry established

Abstract

Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the existence of almost quaternionic structures on 8-dimensional Spinc manifolds and may be of some interest, also, in quaternionic and algebraic geometry.