## Illinois Journal of Mathematics

### Some remarks on the Whitehead asphericity conjecture

S. V. Ivanov

#### 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.

#### Article information

Source
Illinois J. Math., Volume 43, Issue 4 (1999), 793-799.

Dates
First available in Project Euclid: 20 October 2009

https://projecteuclid.org/euclid.ijm/1256060692

Digital Object Identifier
doi:10.1215/ijm/1256060692

Mathematical Reviews number (MathSciNet)
MR1712523

Zentralblatt MATH identifier
0941.57007

#### Citation

Ivanov, S. V. Some remarks on the Whitehead asphericity conjecture. Illinois J. Math. 43 (1999), no. 4, 793--799. doi:10.1215/ijm/1256060692. https://projecteuclid.org/euclid.ijm/1256060692