On the generalized Nielsen realization problem

Abstract

Abstract. The main goal of this paper is to give the first examples of equivariant aspherical Poincaré complexes, that are not realized by group actions on closed aspherical manifolds M. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions. Our examples show that one cannot guarantee that a given action of a finitely generated group π on Euclidean space extends to an action of , a group containing π as a subgroup of finite index, even when all the torsion of lives in π.

Closed aspherical manifolds with center

“…The non-existence of circle actions on T (h)×H will be reduced to the following version of the counterexample to Nielsen: No homology manifold homotopy equivalent to V × H has an involution inducing the same outer automorphism on the fundamental group as h * . This is a stronger version than the one proved in [1], which only dealt with topological manifolds and did not consider the closed hyperbolic manifold factor.…”

“…Relying on [7], Block and Weinberger [1] made use of these elements, with a simpler analysis in the d = 0 case.…”

Bredon–Poincaré duality groups

“…This example is due to Jonathon Block and Schmuel Weinberger and was suggested to us by Jim Davis. Combining Theorems 1.5 and 1.8 of [BW08] gives the following example.…”

Realization problems for diffeomorphism groups

“…Some cases are covered by Schultz [Sch79]. Other instances of Nielsen realization problems for finite subgroups of mapping class groups are discussed by Block-Weinberger in [BW08], who prove several non-realizability results and give a good survey of the state of the art. Examples of non-realizability results for infinite, finitely-generated subgroups for higher-dimensional manifolds are given in [Tsh15]; see Theorem 2.10 below.…”