Aspherical group presentations

Aspherical Labelled Oriented Trees and Knots

“…Following [1], a 2-complex K is called diagrammatically aspherical (DA) if each spherical diagram over P can be converted by diamond moves to one that can be reduced by a folding operation (as in the definition of DR). DR implies DA, which in turn implies aspherical.…”

On the capability of groups

“…For instance, if G/Z is an aspherical group [6] (such as the free abelian group on two generators, or any knot group, or any other one-relator group whose relator is not a proper power) then im{x) = 0 since in this case H^G/Z) = 0. Alternatively, suppose that Z is a direct summand of G; since in this case the surjection G -» G/Z is split, the induced homomorphism H 3 …”

Hereditary Homotopy Equivalences

“…By the equivalence (1) = (1*), a CLA presentation (A": A) is one for which K contracts in a very special manner. The examples in [1] or [5] of aspherical presentations which are not DA, hence not CLA, show that the type of contractibility described by (1*) is restrictive. This is even more transparent in the case of an aspherical presentation for the trivial group.…”

“…In [1], where the Relator Condition is not imposed, Chiswell, Collins, and Huebschmann refer to a group presentation satisfying a version of (1) as CohenLyndon aspherical (CLA), to one satisfying (2) as diagrammatically aspherical (DA), and to one satisfying (3) as aspherical (A). They observe the implications CLA ■» DA -» A (the latter in the presence of the Relator Condition).…”

“…The equivalence (2) = (2*) is established in [5], where it is employed to show that the reverse of the implication DA => A is not valid (a fact also established by an example in [1]). Finally, the equivalence (1) = (1*) and the notion of a hereditary homotopy equivalence are new and form the subject matter for this paper.…”