The homotopy types of $Sp(n)$-gauge groups over $S^{4m}$

TL;DR

This paper investigates the homotopy types of gauge groups associated with principal $Sp(n)$-bundles over spheres, providing bounds on the number of distinct homotopy types for these gauge groups.

Contribution

It offers a partial classification of the homotopy types of $Sp(n)$-gauge groups over spheres, including bounds on their number of homotopy types in specific cases.

Findings

Provides a lower bound for the number of homotopy types of gauge groups.

Gives an upper bound for the homotopy types in special cases ($Sp(3)$ over $S^8$, $Sp(4)$ over $S^{12}$).

Advances understanding of the homotopy classification of gauge groups over spheres.

Abstract

Let $m$ and $n$ be two positive integers such that $m < n$. Denote by $P_{n,k}$ the principal $Sp(n)$-bundle over $S^{4m}$ and $\mathcal{G}_{k,m}(Sp(n))$ be the gauge group of $P_{n,k}$ classified by $k\varepsilon'$, where $\varepsilon'$ is a generator of $\pi_{4m}(B(Sp(n)))\cong\mathbb{Z}$. In this article, we will partially classify the homotopy types of $\mathcal{G}_{k,m}(Sp(n))$ by giving a lower bound for the number of homotopy types of $\mathcal{G}_{k,m}(Sp(n))$. Also, in special cases $Sp(3)$-gauge groups over $S^8$ and $Sp(4)$-gauge groups over $S^{12}$ we give an upper bound for the number of homotopy types of these gauge groups.