## Abstract

The Whitehead asphericity conjecture claims that if 〈script A sign∥ℛ〉 is an aspherical group presentation, then for every S ⊂ ℛ the subpresentation 〈A∥script S sign〉 is also aspherical. It is proven that if the Whitehead conjecture is false then there is an aspherical presentation E = 〈script A sign∥ℛ ∪ {z}〉 of the trivial group E, where the alphabet sript A sign is finite or countably infinite and z ∈ script A, sign such that its subpresentation 〈script A sign∥ℛ〉 is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite script A sign and ℛ) then there is a finite aspherical presentation 〈script A sign∥ℛ〉, ℛ = {R_{1}, R_{2}, ..., R_{n}}, such that for every scipt S sign ⊆ ℛ the subpresentation 〈script A sign∥script S sign〉 is aspherical and the subpresentation 〈script A sign∥R_{1} R_{2}, ..., R_{n}〉 of aspherical 〈script A sign∥R_{1} R_{2}, R_{2}, R_{3}, ..., R_{n}〉 is not aspherical.

Original language | English (US) |
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Pages (from-to) | 793-799 |

Number of pages | 7 |

Journal | Illinois Journal of Mathematics |

Volume | 43 |

Issue number | 4 |

DOIs | |

State | Published - 1999 |

## ASJC Scopus subject areas

- General Mathematics