Topology of moduli of parabolic connections with fixed determinant

Let $X$ be a compact Riemann surface of genus $g \geq 2$ 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, $\mathcal{M}_{\mathrm{conn}}(\alpha, r, \xi)$) denote the moduli space of stable parabolic bundles (respectively, parabolic connections) of rank $r$ $(\geq 2)$, determinant $\xi$ and full flag generic rational parabolic weight type $\alpha$. We show that $ \pi_k(\mathcal{M}_{\mathrm{conn}}(\alpha, r, \xi)) \cong \pi_k(\mathcal{M}(\alpha, r, \xi)) $ for $k \leq2(r-1)(g-1)-1$. As a consequence, we deduce that the moduli space $\mathcal{M}_{\mathrm{conn}}(\alpha, r, \xi)$ is simply connected. We also show that the Hodge structures on the torsion-free parts of both the cohomologies $H^k(\mathcal{M}_{\mathrm{conn}}(\alpha, r, \xi),\mathbb{Z})$ and $H^k(\mathcal{M}(\alpha, r, \xi),\mathbb{Z})$ are isomorphic for all $k\leq 2(r-1)(g-1)+1$.