The homotopy types of $SU(n)$-gauge groups over $S^{2m}$

Let $m$ and $n$ be two positive integers such that $m \leqslant n$. Denote by $P_{n,k}$ the principal $SU(n)$-bundle over $S^{2m}$ with Chern class $c_m (P_{n,k}) = (m-1)!k$ and let $\mathcal{G}_{k,m} (SU(n))$ be the gauge group of $P_{n,k}$ classified by $k \varepsilon^\prime$, where $\varepsilon^\prime$ is a generator of $\pi_{2m} (B(SU(n)) \cong \mathbb{Z}$. In this article we partially classify the homotopy types of $\mathcal{G}_{k,m} (SU(n))$, by showing that if there is a homotopy equivalence $\mathcal{G}_{k,m} (SU(n)) \simeq \mathcal{G}_{k^\prime,m} (SU(n))$ then in case $m$ is odd and $m \geqslant 3, (\frac{2}{(m-1)!} p_2, k) = (\frac{2}{(m-1)!}p_2, k^\prime)$ and in case $m$ is even and $m \geqslant 4, (\frac{1}{2(m-1)!} p_2, k) = (\frac{1}{2(m-1)!} p_2, k^\prime)$, where $p_2 = (n+2)(n+1) n (n-1) \dotsm (n-m+2)$. We study the group $[\Sigma^{2n} \mathbb{C}P^{n-1}, SU(n)]$. Also we discuss the order of the Samelson product $S^{2m-1} \wedge \Sigma \mathbb{C} P^{n-1} \to SU(n)$ when $m \lt n$.

Keywords

gauge group, homotopy type, Lie group, homotopy equivalence

2010 Mathematics Subject Classification

Primary 55P15. Secondary 54C35.

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In memory of Professor Mohammad Ali Asadi-Golmankhaneh.

Received 8 December 2020

Received revised 17 January 2021

Accepted 16 February 2021

Published 30 March 2022