Local indicability in the presence of diagrammatic reducibility

TL;DR

The paper addresses when local indicability is inherited by a subcomplex inside a diagrammatically reducible 2-complex, with implications for Whitehead's asphericity conjecture for labeled oriented trees (LOTs). It develops a Corson-Trace–style criterion for diagrammatic reducibility away from a subcomplex and derives a relative theorem that transfers DR and local indicability from a target complex $Y$ to a source complex $X$ via an immersion outside a subcomplex. A key contribution is showing that if $f:X\to Y$ is an immersion outside $L$, $Y$ is DR away from $f(L)$, and $L$ is DR with $\pi_1$ locally indicable on each component (and $\pi_1(Y)$ locally indicable), then $\pi_1(X)$ is locally indicable; in particular, reduced injective LOTs that are DR(2) and whose quotients are DR(2) are locally indicable. The results connect to non-positive immersions, vertex asphericity (VA), and Whitehead's conjecture, providing concrete criteria to establish LOT local indicability and, by extension, LOT asphericity.

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.

Theorems & Definitions (37)

• Theorem 2.1
• Theorem 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 2.6
• proof
• Theorem 2.7
• proof
• Theorem 2.8