Some remarks on the Whitehead asphericity conjecture

The Whitehead asphericity conjecture claims that if (A\\R) is an aspherical group presentation, then for every S subset of R the subpresentation (A\\S) is also aspherical. It is proven that if the Whitehead conjecture is false then there is an aspherical presentation E = [A\\R U {z}] of the trivial group E, where the alphabet A is finite or countably infinite and z is an element of A, such that its subpresentation [A\\R] is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite A and R) then there is a finite aspherical presentation [A\\R], R = {R-1, R-2, ..., R-n}, such that for every S subset of or equal to R the subpresentation (A\\S) is aspherical and the subpresentation [A\\R1R2, R-3, ..., R-n] of aspherical [A\\R-1 R-2, R-2, R-3, ..., R-n] is not aspherical.