The Homotopy Types of $SU(4)$-Gauge Groups

Let $\mathcal{G}_k$ be the gauge group of the principal $SU(4)$-bundle over $S^4$ with second Chern class $k$ and let $p$ be a prime. We show that there is a rational or $p$-local homotopy equivalence $Ω\mathcal{G}_k\simeqΩ\mathcal{G}_{k'}$ if and only if $(60,k)=(60,k')$.

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