Homotopy types of gauge groups of $\mathrm{PU}(p)$-bundles over spheres

Simon Rea

Published 2020-04-10Version 1

We examine the relation between the gauge groups of $\mathrm{SU}(n)$- and $\mathrm{PU}(n)$-bundles over $S^{2i}$, with $2\leq i\leq n$, particularly when $n$ is a prime. As special cases, for $\mathrm{PU}(5)$-bundles over $S^4$, we show that there is a rational or $p$-local equivalence $\mathcal{G}_k\simeq_{(p)}\mathcal{G}_l$ for any prime $p$ if, and only if, $(120,k)=(120,l)$, while for $\mathrm{PU}(3)$-bundles over $S^6$ there is an integral equivalence $\mathcal{G}_k\simeq\mathcal{G}_l$ if, and only if, $(120,k)=(120,l)$.