Topology of moduli of parabolic connections with fixed determinant

Let $X$ be a compact Riemann surface of genus $g\ge 2$ and $D\subset X$ be a fixed finite subset. Let $\xi$ be a line bundle of degree $d$ over $X$. Let $\mathcal{M}(\alpha ,r,\xi )$ (respectively, ${\mathcal{M}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}(\alpha ,r,\xi )$) denote the moduli space of stable parabolic bundles (respectively, parabolic connections) of rank $r$ $(\ge 2)$, determinant $\xi$ and full flag generic rational parabolic weight type $\alpha$. We show that ${\pi }_k({\mathcal{M}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}(\alpha ,r,\xi ))\cong {\pi }_k(\mathcal{M}(\alpha ,r,\xi ))$ for $k\le 2(r-1)(g-1)-1$. As a consequence, we deduce that the moduli space ${\mathcal{M}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}(\alpha ,r,\xi )$ is simply connected. We also show that the Hodge structures on the torsion-free parts of both the cohomologies $H^k({\mathcal{M}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}(\alpha ,r,\xi ),\mathbb{Z})$ and $H^k(\mathcal{M}(\alpha ,r,\xi ),\mathbb{Z})$ are isomorphic for all $k\le 2(r-1)(g-1)+1$.