Finiteness of $A_n$-equivalence types of gauge groups

Let $B$ be a finite CW complex and $G$ a compact connected Lie group. We show that the number of gauge groups of principal $G$-bundles over $B$ is finite up to $A_n$-equivalence for $n<\infty$. As an example, we give a lower bound of the number of $A_n$-equivalence types of gauge groups of principal $\SU(2)$-bundles over $S^4$.

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