TY - JOUR
T1 - Some remarks on the whitehead asphericity conjecture
AU - Ivanov, S. V.
N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 1999
Y1 - 1999
N2 - 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.
AB - 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.
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U2 - 10.1215/ijm/1256060692
DO - 10.1215/ijm/1256060692
M3 - Article
AN - SCOPUS:0040111773
SN - 0019-2082
VL - 43
SP - 793
EP - 799
JO - Illinois Journal of Mathematics
JF - Illinois Journal of Mathematics
IS - 4
ER -