On the non-realizability of braid groups by diffeomorphisms (Q2812001)

Let \(S\) be a compact surface, \(\mathrm{Diff}(S)\) the group of orientation preserving \(C^1\) diffeomorphisms of \(S\). \(\mathrm{Mod}(S)=\pi_0(\mathrm{Diff}(S))\) the mapping class group, \(p:\mathrm{Diff}(S)\to\mathrm{Mod}(S)\) standard projection. Morita proved that for surfaces of genus greater than 4, \(\mathrm{Mod}(S)\) cannot be realized as a subgroup of \(\mathrm{Diff}(S)\), i.e., \(p\) has no isomorphic section. Here we start from the surface braid group \(B_n(S)=\pi_1(\mathrm{Conf}_n(S))\), \(\mathrm{Conf}_n(S)\) is the configuration space of \(n\)-tuples of distinct, unordered points of \(S\) and a map \(\mathcal{P}:B_n(S) \to\mathrm{Mod}(S,n)\) which is a generalization of point pushing and consider its realizability. Namely, one takes into account action of \(\mathrm{Diff}(S)\) onto \(\mathrm{Conf}_n(S)\). It acts on \(\mathrm{Conf}_n(S)\) with the stabilizer of \([\phi]\) giving \(\mathcal{P}:\pi_1(\mathrm{Conf}_n(S))\to \pi_0\mathrm{Diff}(S,[\phi])\). The authors define: \(\mathcal{P}\) is realized by \(C^1\) diffeomorphism if \(\mathcal{P}\) can be factored through \(\mathrm{Diff}(S)\) via homomorphism \(\sigma:\pi_1(\mathrm{Conf}_n(S))\to\mathrm{Diff}(S,[\phi])\). The obtained result is the following: Let \(S\) be a compact surface. If \(\partial{S}=\emptyset\), then \(\mathcal{P}\) is not realized for \(n\geq 6\). In the case \(\partial{S} \neq \emptyset\), then for \(n\geq 5\). Beside this the used technique is applied to some more general situations.