Nielsen realization for 3-manifolds containing two-sided projective planes (Q1566431)

Let \(N\) be a compact manifold, denote by \(h:\text{Homeo}(N)\to {\mathcal M}(N)\) the canonical quotient from the group of homeomorphisms of \(N\) to the mapping class group of \(N\) and by \(\beta:{\mathcal M}(N)\to \text{Out}_{\partial}(\pi_1(N))\) the natural homomorphism from \({\mathcal M}(N)\) to the group of the outer automorphisms of \(\pi_1(N)\) which preserve the peripheral structure. Suppose that \(\psi: G\to\text{Out}_{\partial}(\pi_1(N))\) is a given homomorphism from some finite group \(G\). The author considers the following realization problem: does there exist an injection \(\xi:G\to\text{Homeo}(N)\) such that \(\psi=\beta h\xi\)? If \(\psi\) is an injection and \(\beta\) is an isomorphism, then this is the classical Nielsen Realization problem, which gives a way to translate the algebra of automorphisms of \(\pi_1(N)\) to the topology of homeomorphisms of \(N\). This problem has been already solved in a number of cases: by Kerckhoff in dimension two, by Heil and Tollefson for compact, orientable, irreducible, sufficiently large 3-manifolds, when \(G={\mathbb Z}_n\), by Zieschang and Zimmermann for Seifert fibered 3-manifolds (generalized by Zimmermann to Haken manifolds). In this paper, the author proves some realization results for 3-manifolds containing two-sided projective planes, which are therefore non-orientable.