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Winter 1999 Some remarks on the Whitehead asphericity conjecture
S. V. Ivanov
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Illinois J. Math. 43(4): 793-799 (Winter 1999). DOI: 10.1215/ijm/1256060692

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.

Citation

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S. V. Ivanov. "Some remarks on the Whitehead asphericity conjecture." Illinois J. Math. 43 (4) 793 - 799, Winter 1999. https://doi.org/10.1215/ijm/1256060692

Information

Published: Winter 1999
First available in Project Euclid: 20 October 2009

zbMATH: 0941.57007
MathSciNet: MR1712523
Digital Object Identifier: 10.1215/ijm/1256060692

Subjects:
Primary: 57M20
Secondary: 20F05, 20F06

Rights: Copyright © 1999 University of Illinois at Urbana-Champaign

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Vol.43 • No. 4 • Winter 1999
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