Local connectivity of spaces of finite group actions on 3-manifolds
The author shows that the space \({\mathcal A}_{PL}(G,M)\) of PL-actions of a finite group G on a closed 3-manifold M with the compact-open topology is strongly locally connected. That is, for each point \(\phi\in {\mathcal A}_{PL}(G,M)\) and for every neighborhood V of \(id_ M\) in the space \({\mathcal H}_{PL}(M)\) of PL-homeomorphisms of M with the compact-open topology there exists a neighborhood U of \(\phi\) in \({\mathcal A}_{PL}(G,M)\) such that for every \(\psi\in U\) there exists a PL-isotopy \(h_ t\in V\) such that \(h_ 0=id_ M\) and \(h_ 1\) conjugates \(\psi\) to \(\phi\) \((\psi =h_ 1\phi h_ 1^{-1})\). In particular, \({\mathcal A}_{PL}(G,M)\) is locally arc-wise connected and close PL-G-actions are equivalent. The author uses the Edwards-Kirby deformation theorem, the equivariant Dehn lemma and the generalized Smith conjecture (it was proved by \textit{W. Thurston} but has not been published yet. One can be refered to Thurston's announcement [''Three-manifolds with symmetry'', preprint (January, 1982)] and to the course given by C. Hodgson at Columbia University, 1986-1987).
space of PL-actions of a finite group on a closed 3-manifold
scientific article; zbMATH DE number 3998798
scientific article