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 $\alpha \text{\hspace{0.17em}⁣}:X\to BG$. 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 $n\le 4$. As an application, we classify the homotopy types of $SO(n)$-gauge groups over a Riemann surface for $n\le 4$.