Finiteness conditions in covers of Poincar\'e duality spaces

TL;DR

This paper characterizes when a finite Poincaré duality 4-space has a finitely dominated infinite cover, linking it to universal covers or specific mapping tori structures.

Contribution

It provides a precise criterion for finiteness conditions in covers of Poincaré duality spaces, extending understanding of their topological and homotopy properties.

Findings

Finitely dominated covers relate to universal covers being finitely dominated.

Finitely covered spaces are characterized as mapping tori of PD3-complexes.

The results unify conditions for finiteness in 4-dimensional Poincaré duality spaces.

Abstract

A closed 4-manifold (or, more generally, a finite $PD_4$-space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus of a self homotopy equivalence of a $PD_3$-complex.