$PD_3$-complexes bound

Jonathan A. Hillman

$PD_3$-complexes bound

Geometric Topology 2023-01-18 v5

Authors: Jonathan A. Hillman

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.

Keywords

manifolds homotopy groups

Cite

@article{arxiv.2109.09947,
  title  = {$PD_3$-complexes bound},
  author = {Jonathan A. Hillman},
  journal= {arXiv preprint arXiv:2109.09947},
  year   = {2023}
}

Comments

v2: Proof of Theorem 1 elaborated. New Theorem 2, on $PD_3$-complexes with manifold 1-skeleta; the original \S3 now an appendix. v3: new section, extending a theorem of Daverman to the $PD_4$-context. v4: appendix deleted, \S4 moved to follow \S1. Other minor changes. v5: exposition of Theorem 1, Lemma 3 and Theorem 4 tightened

$PD_3$-complexes bound

Abstract

Keywords

Cite

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