Some remarks on the whitehead asphericity conjecture

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, ..., 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∥R₁ R₂, ..., R_n〉 of aspherical 〈script A sign∥R₁ R₂, R₂, R₃, ..., R_n〉 is not aspherical.