A topological approach to the Nielsen's realization problem for Haken 3-manifolds (Q2726679)

The mapping class group \({\mathfrak M}(M)\) of a manifold \(M\) is the quotient group of the group of auto-homeomorphisms of \(M\) by the subgroup of those isotopic to the identity. If \(c\in{\mathfrak M}(M)\) has finite order \(n\), and there is an autohomeomorphism \(f\) in \(c\) of order \(n\), \(c\) is said to be realized by \(f\). If a finite subgroup \(G\) of \({\mathfrak M}(M)\) is isomorphic to a group \(F\) of autohomeomorphisms by the function mapping each element of \(F\) to its class, \(G\) is said to be realized by \(F\). Not all finite subgroups of \({\mathfrak M}(M)\) can be realized. If \(F_{2p+1}\) is an orientable closed surface with genus \(2p+1\), \(p>0\), there is a finite subgroup of \({\mathfrak M}(F_{2p+1} \times S^1)\) which cannot be realized (Zieschang and Zimmermann). The main point of this paper is to give a completely topological proof, avoiding arguments about algebraic obstruction, of the following result of Zimmermann: Let \(M\) be a Haken manifold such that \(\partial M\) is either empty or a union of tori and \(M\) is not a closed Seifert manifold. Then mapping classes of finite order are realized by autohomeomorphisms of that order. The paper builds by showing versions of this result for a class of hyperbolic 3-manifolds, Seifert 3-manifolds, and finally Haken 3-manifolds.