Local indicability in the presence of diagrammatic reducibility

TL;DR

This paper explores conditions under which local indicability of fundamental groups is preserved in diagrammatically reducible 2-complexes, with implications for Whitehead's asphericity conjecture and properties of labeled oriented trees.

Contribution

It introduces a stronger Corson-Trace like characterization of diagrammatic reducibility away from a subcomplex and applies it to prove local indicability for certain labeled oriented trees.

Findings

Local indicability is preserved under certain diagrammatic reducibility conditions.

Injective labeled oriented trees of degree 2 are locally indicable.

Quotients of such labeled oriented trees are also locally indicable.

Abstract

If a complex $X$ is a subcomplex of a diagrammatically reducible 2-complex $Y$ that has locally indicable fundamental group, then $X$ has locally indicable fundamental group. This is a consequence of the Corson-Trace characterization of diagrammatic reducibility. In this paper we use a Corson-Trace like characterization of diagrammatic reducibility away from a subcomplex to obtain a considerable stronger result. We apply this to the question of local indicability in the context of Whitehead's asphericity conjecture. We show that an injective labeled oriented tree (LOT) that is diagrammatically reducible of degree 2, and all its quotients are as well, is locally indicable.