Relative Vertex Asphericity

TL;DR

This paper introduces tests for a relative version of vertex asphericity, enabling the detection of asphericity in pairs of 2-complexes, with applications to labeled oriented trees.

Contribution

It develops a relative weight test for vertex asphericity applicable to pairs of 2-complexes, simplifying previous proofs and extending the theory.

Findings

Relative weight test applies to injective labeled oriented trees

Labeled oriented trees are shown to be VA and aspherical

Simplifies previous proofs of asphericity results

Abstract

Diagrammatic reducibility DR and its generalization vertex asphericity VA are combinatorial tools developed for detecting asphericity of a 2-complex. Here we present tests for a relative version of VA that apply to pairs of 2-complexes $(L,K)$, where $K$ is a subcomplex of $L$. We show that a relative weight test holds for injective labeled oriented trees, implying that they are VA and hence aspherical. This strengthens a result obtained by the authors in 2017 and simplifies the original proof.