On the homotopy types of $\mathrm{Sp}(n)$ gauge groups

TL;DR

This paper refines the classification of homotopy types of gauge groups of principal Sp(n)-bundles over S^4, relating it to Samelson products and classifying p-local types under specific conditions.

Contribution

It improves previous results on the homotopy types of Sp(n) gauge groups and connects these types with Samelson products, providing a detailed p-local classification.

Findings

Refined the homotopy classification of gauge groups of Sp(n) bundles.

Connected homotopy types with the order of Samelson products in Sp(n).

Classified p-local homotopy types for certain primes and n.

Abstract

Let $\mathcal{G}_{k,n}$ be the gauge group of the principal $\mathrm{Sp}(n)$-bundle over $S^4$ corresponding to $k\in\mathbb{Z}\cong\pi_3(\mathrm{Sp}(n))$. We refine the result of Sutherland on the homotopy types of $\mathcal{G}_{k,n}$ and relate it with the order of a certain Samelson product in $\mathrm{Sp}(n)$. Then we classify the $p$-local homotopy types of $\mathcal{G}_{k,n}$ for $(p-1)^2+1\ge 2n$.