Homotopy of gauge groups over non-simply-connected five dimensional manifolds

TL;DR

This paper investigates the homotopy properties of gauge groups over non-simply-connected 5-manifolds, providing decompositions, classifications, and periodicity results that extend to rational and higher-dimensional cases.

Contribution

It offers new homotopy decompositions and classification results for gauge groups over certain 5-manifolds, including periodicity and rational case analyses.

Findings

Homotopy decompositions of gauge groups based on manifold structures

Partial classification of gauge groups over non-simply-connected 5-manifolds

Periodic homotopy group results analogous to Bott periodicity

Abstract

Both the gauge groups and $5$-manifolds are important in physics and mathematics. In this paper, we combine them together to study the homotopy aspects of gauge groups over $5$-manifolds. For principal bundles over non-simply connected oriented closed $5$-manifolds of certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to Bott periodicity. Our treatments here are also very effective for rational gauge groups in general context, and applicable for higher dimensional manifolds.