The homotopy types of $Sp(n)$-gauge groups over $\mathbb{C}P^2$

Let $n \gt 2$ and $\mathcal{G}_k (\mathbb{C}P^2)$ be the gauge groups of the principal $Sp(n)$-bundles over $\mathbb{C}P^2$. In this article we partially classify the homotopy types of $\mathcal{G}_k (\mathbb{C}P^2)$ by showing that if there is a homotopy equivalence $\mathcal{G}_k (\mathbb{C}P^2) \simeq \mathcal{G}_{k^\prime} (\mathbb{C}P^2)$ then $(k, 4n(2n + 1)) = (k^\prime , 4n(2n + 1))$.

Keywords

gauge group, homotopy type, symplectic group

2010 Mathematics Subject Classification

Primary 55P15. Secondary 54C35.

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In memory of Professor Mohammad Ali Asadi-Golmankhaneh.

Received 19 December 2021

Received revised 22 April 2022

Accepted 28 April 2022

Published 12 April 2023