Equivariant Poincar\'e duality for cyclic groups of prime order and the Nielsen realisation problem

TL;DR

This paper applies equivariant Poincaré duality to cyclic groups of prime order to address a technical condition in the Nielsen realisation problem, also providing a new characterization of Poincaré spaces and a refined duality concept.

Contribution

It extends equivariant Poincaré duality theory to cyclic groups of prime order and offers a complete characterization of C_p-Poincaré spaces, advancing the understanding of duality in equivariant topology.

Findings

Removed a technical condition in Nielsen realisation for aspherical manifolds.

Characterized C_p-Poincaré spaces completely.

Introduced a refined notion of virtual Poincaré duality groups.

Abstract

In this companion article to [HKK24], we apply the theory of equivariant Poincar\'e duality developed there in the special case of cyclic groups $C_p$ of prime order to remove, in a special case, a technical condition given by Davis--L\"uck [DL24] in their work on the Nielsen realisation problem for aspherical manifolds. Along the way, we will also give a complete characterisation of $C_p$--Poincar\'e spaces as well as introduce a genuine equivariant refinement of the classical notion of virtual Poincar\'e duality groups which might be of independent interest.