Some remarks on the Whitehead asphericity conjecture
The Whitehead asphericity conjecture claims that every subcomplex of an aspherical 2-complex (trivial second homotopy group) is aspherical; in group-theoretical terms, it states that any subpresentation (with the same set of generators) of an aspherical group presentation is aspherical (i.e. the canonical 2-complex associated to the group presentation is aspherical). Using ideas of small cancellation theory, it is proved in the present paper that if the Whitehead asphericity conjecture is false then there is a counterexample which arises by deleting from an aspherical presentation of the trivial group a single relator which is just one of the generators. Here the generating set is finite or countable; it can be chosen finite if there is a finite presentation giving a counterexample to the Whitehead asphericity conjecture. Also, it is proved that any finite aspherical presentation is Andrews-Curtis equivalent to a finite aspherical presentation with the property that also every subpresentation is aspherical; as a consequence, every finite aspherical 2-complex can be 3-deformed to a finite 2-complex all of whose subcomplexes are aspherical.
scientific article; zbMATH DE number 1361058