$PD_3$-complexes bound

TL;DR

This paper proves that every $PD_3$-complex bounds a $PD_4$-pair, with additional results relating to orientability, manifold skeletons, and fundamental groups, advancing the understanding of Poincaré duality complexes in topology.

Contribution

It establishes that all $PD_3$-complexes bound $PD_4$-pairs and explores conditions under which these complexes relate to 3-manifolds and free groups.

Findings

Every $PD_3$-complex bounds a $PD_4$-pair.

If $P$ is orientable, then $ au_1(Z)=1$.

$P$ with a manifold 1-skeleton is homotopy equivalent to a closed 3-manifold.

Abstract

We show that every $PD_3$-complex $P$ bounds a $PD_4$-pair $(Z,P)$. If $P$ is orientable we may assume that $\pi_1(Z)=1$. We show also that if $P$ has a manifold 1-skeleton then it is homotopy equivalent to a closed 3-manifold, and that if the inclusion of $Z$ into $P$ induces an isomorphism on fundamental groups then $\pi_1(Z)$ is a free group.