On real moduli spaces over M-curves

Let $F$ be a genus $g$ curve and $\sigma :F\to F$ a real structure with the maximal possible number of fixed circles. We study the real moduli space $\N’ = \Fix (\sigma^{#})$where$\sigma^{#}: \N \to \N$ is the induced real structure on the moduli space $\mathbb{N}$ of stable holomorphic bundles of rank 2 over $F$ with fixed non-trivial determinant. In particular, we calculate $H^* (\N’,\mathbb Z)$inthecaseof$g = 2$,generalizingThaddeu{s}^{\mathrm{\prime }}approachtocomputing$H^* (\N,\mathbb Z)$.