$PD_3$-complexes bound

We show that every $PD_3$-complex $P$ bounds a $PD_4$-pair $(Z,P)$. If $P$ is orientable we may assume that $π_1(Z)=1$. 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 $π_1(Z)$ is a free group.

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