The Local Structure of Injective LOT-Complexes

TL;DR

This paper characterizes the local structure of injective LOT complexes, showing they are aspherical and their links are bi-forests under certain conditions, advancing understanding of Whitehead's asphericity conjecture.

Contribution

It provides a complete description of the link of reduced injective LOT complexes and proves their asphericity and local indicability of their fundamental groups.

Findings

Links of reduced injective LOT complexes are bi-forests.

Injective LOT complexes are aspherical.

Fundamental groups of these complexes are locally indicable.

Abstract

Labeled oriented trees, LOT's, encode spines of ribbon discs in the 4-ball and ribbon 2-knots in the 4-sphere. The unresolved asphericity question for these spines is a major test case for Whitehead's asphericity conjecture. In this paper we give a complete description of the link of a reduced injective LOT complex. An important case is the following: If $\Gamma$ is a reduced injective LOT that does not contain boundary reduced sub-LOTs, then $lk(K(\Gamma))$ is a bi-forest. As a consequence $K(\Gamma)$ is aspherical, in fact DR, and its fundamental group is locally indicable. We also show that a general injective LOT complex is aspherical. Some of our results have already appeared in print over the last two decades and are collected here.