Indecomposable $PD_3$-complexes

TL;DR

This paper classifies indecomposable $PD_3$-complexes with certain fundamental groups, showing they are orientable with a tree-like structure and specific group properties, and discusses their realization and related open questions.

Contribution

It provides a classification of indecomposable $PD_3$-complexes based on their fundamental groups and proposes a strategy for a major open problem in topology.

Findings

Indecomposable $PD_3$-complexes are orientable with a tree structure.

Edge groups are $Z/2Z$, and most vertex groups are dihedral of order $2m$ with $m$ odd.

Every such group can be realized by some $PD_3$-complex.

Abstract

We show that if $X$ is an indecomposable $PD_3$-complex and $\pi_1(X) is the fundamental group of a reduced finite graph of finite groups but is not virtually cyclic then $X$ is orientable, the underlying graph is a tree, all the edge groups are $Z/2Z$ and all but at most one of the vertex groups is dihedral of order $2m$ with $m$ odd. Every such group is realized by some $PD_3$-complex. We also propose a strategy for tackling the question of whether every $PD_3$-complex has a finite covering space which is homotopy equivalent to a closed orientable 3-manifold.