On the cohomology of the classifying spaces of $SO(n)$-gauge groups over $S^2$

Let $\mathcal{G}_\alpha (X,G)$ be the $G$-gauge group over a space $X$ corresponding to a map $\alpha : X \to B\mathcal{G}_1$. We compute the integral cohomology of $B\mathcal{G}_1 (S^2, SO(n))$ for $n = 3, 4$. We also show that the homology of $B\mathcal{G}_1 (S^2, SO(n))$ is torsion free if and only if $n \leqslant 4$. As an application, we classify the homotopy types of $SO(n)$-gauge groups over a Riemann surface for $n \leqslant 4$.

Keywords

cohomology, gauge group, classifying space

2010 Mathematics Subject Classification

55R40

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Received 6 May 2023

Received revised 28 July 2023

Accepted 1 August 2023

Published 18 September 2024