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

Sajjad Mohammadi

Published 2022-04-19Version 1

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 partially classify the homotopy types of $\mathcal{G}_{k,m}(Sp(n))$. Also we will study the order of the Samelson product $S^{4m-1}\wedge Q_{n-m+1}\rightarrow Sp(n)$, where $Q_{n-m+1}$ be the symplectic quasi-projective space of rank $n-m+1$.