Non-realizability of some big mapping class groups (Q6602175)

For an orientable surface \(X\) let \(\mathcal H^+(X)\) and \(\mathcal M(X)\) denote, respectively, the group of orientation-preserving homeomorphisms and the mapping class group of \(X\), and \(p:\mathcal H^+(X)\to\mathcal M(X)\) the forgetful homomorphism: a subscript \(c\) will denote `with compact support'. It is shown that \(p:\mathcal H^+(S_{3,1})\to\mathcal M(S_{3,1})\) has no sections, where \(S_{3,1}\) is the surface of genus 3 and 1 boundary component, hence \(p_c:\mathcal H^+_c(X)\to\mathcal M_c(X)\) has no sections when the surface \(X\) has a genus 3 subsurface. Also if \(X=\mathbb R^2\setminus C\) or \(X=\mathbb S^2\setminus C\) for certain \(C\), including the Cantor set and the ordinal \(\omega_\alpha+1\), then \(p:\mathcal H^+(X)\to\mathcal M(X)\) has no sections.