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Let $G$ be a compact connected Lie group with $\pi_1(G)\cong\mathbb{Z}$. Westudy the homotopy types of gauge groups of principal $G$-bundles over Riemannsurfaces. This can be applied to an explicit computation of the homotopy groupsof the moduli spaces of stable vector bundles over Riemann surfaces.

Homotopy types of gauge groups over Riemann surfaces