The space of immersed surfaces in a manifold (Q2842337)

Let \(M\) be a smooth manifold, not necessarily compact and possibly with boundary, and let \(\dot M\) denote its interior. For a closed orientable surface \(\Sigma_g\) of genus \(g\), denote by \(\text{Imm}(\Sigma_g, \dot M)\) the space of immersions of \(\Sigma_g\) into the interior of \(M\), equipped with the Whitney \(C^{\infty}\)-topology. The group \(\text{Diff}^{+}(\Sigma_g)\) of orientation preserving diffeomorphisms of \(\Sigma_g\) is also equipped with the Whitney \(C^{\infty}\)-topology, and acts continuously on \(\text{Imm}(\Sigma_g, \dot M)\) by precomposition of functions. The \textsl{space of immersed surfaces of genus \(g\) in \(M\)} is defined to be the quotient space \(\mathcal{I}_g(M) := \text{Imm}(\Sigma_g, \dot M)/\text{Diff}^{+}(\Sigma_g)\). Accordingly, a point in \(\mathcal{I}_g(M)\) is represented by an unparametrized immersed oriented surface of genus \(g\) into the interior of \(M\).NEWLINENEWLINEThe author makes a contribution to the understanding of the differential topology of the space \(\mathcal{I}_g(M)\) by computing its rational cohomology in a stable range which goes to infinity with \(g\). Using deep work of \textit{S. Galatius} et al., [Acta Math. 202, No. 2, 195--239 (2009; Zbl 1221.57039)] the cohomology is identified with that of certain interesting infinite loop spaces.