The Homotopy Types of $SU(4)$-Gauge Groups

TL;DR

This paper investigates the homotopy types of gauge groups of principal $SU(4)$-bundles over $S^4$, revealing conditions under which their loop spaces are homotopy equivalent after rational or p-localization.

Contribution

It establishes a precise classification of the homotopy types of these gauge groups based on the second Chern class and prime localization.

Findings

Homotopy equivalence of looped gauge groups depends on the second Chern class and prime localization.

Classification criterion involves the greatest common divisor with 60.

Provides a complete description of the homotopy types for these gauge groups.

Abstract

Let $\mathcal{G}_k$ be the gauge group of the principal $SU(4)$-bundle over $S^4$ with second Chern class $k$ and let $p$ be a prime. We show that there is a rational or $p$-local homotopy equivalence $\Omega\mathcal{G}_k\simeq\Omega\mathcal{G}_{k'}$ if and only if $(60,k)=(60,k')$.