Some remarks on the Whitehead asphericity conjecture

Abstract

The Whitehead asphericity conjecture claims that if $\langle\mathcal{A}||\mathcal{R}\rangle$ is an aspherical group presentation, then for every $\mathcal{S} \subset \mathcal{R}$ the subpresentation $\langle\mathcal{A}||\mathcal{S}\rangle$ is also aspherical. It is proven that if the Whitehead conjecture is false then there is an aspherical presentation $E = \langle\mathcal{A}||\mathcal{R} \cup \{z\}\rangle$ of the trivial group $E$, where the alphabet $\mathcal{A}$ is finite or countably infinite and $z \in \mathcal{A}$, such that its subpresentation $\langle\mathcal{A}||\mathcal{R}\rangle$ is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite $\mathcal{A}$ and $\mathcal{R}$) then there is a finite aspherical presentation $\langle\mathcal{A}||\mathcal{R}\rangle$, $\mathcal{R} = \{R_{1},R_{2},\ldots,R_{n}}$, such that for every $\mathcal{S} \subseteq \mathcal{R}$ the subpresentation $\langle\mathcal{A}||\mathcal{S}\rangle$ is aspherical and the subpresentation $\langle\mathcal{A}||R_{1}R_{2},R_{3},\ldots,R_{n}$ of aspherical $\langle\mathcal{A}||R_{1}R_{2},R_{3},\ldots,R_{n}$ is not aspherical.