Some remarks on the whitehead asphericity conjecture

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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 = {R1, R2, ..., Rn}, such that for every scipt S sign ⊆ ℛ the subpresentation 〈script A sign∥script S sign〉 is aspherical and the subpresentation 〈script A sign∥R1 R2, ..., Rn〉 of aspherical 〈script A sign∥R1 R2, R2, R3, ..., Rn〉 is not aspherical.

Original languageEnglish (US)
Pages (from-to)793-799
Number of pages7
JournalIllinois Journal of Mathematics
Issue number4
StatePublished - 1999

ASJC Scopus subject areas

  • Mathematics(all)


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