An answer to the Whitehead asphericity question

TL;DR

This paper proves that for aspherical presentations of the trivial group, removing a relation preserves asphericity, thus positively resolving the Whitehead asphericity problem in this context.

Contribution

It provides a positive answer to the Whitehead asphericity question for trivial group presentations by showing subpresentations remain aspherical.

Findings

Subpresentations of aspherical trivial group presentations are also aspherical.

The result confirms the Whitehead asphericity problem for a class of presentations.

The proof advances understanding of combinatorial group theory and asphericity properties.

Abstract

The Whitehead asphericity problem, regarded as a problem of combinatorial group theory, asks whether any subpresentation of an aspherical group presentation is also aspherical. We give a positive answer to this question by proving that if $\cP=(\mathbf{x}, \mathbf{r})$ is an aspherical presentation of the trivial group, and $r_{0} \in \mathbf{r}$ a fixed relation, then $\cP_{1}=(\mathbf{x}, \mathbf{r}_{1})$ is aspherical where $\mathbf{r}_{1}=\mathbf{r} \setminus \{r_{0}\}$.