Asphericity for certain groups of cohomological dimension 2

TL;DR

This paper characterizes asphericity of certain 2-complexes and groups of cohomological dimension 2, linking algebraic properties of modules over group rings to topological asphericity.

Contribution

It establishes a criterion for asphericity of 2-complexes and groups based on the vanishing of specific homology subgroups and module properties, connecting algebraic and topological aspects.

Findings

A 2-complex is aspherical iff its spherical 2-cycles subgroup is zero.

Subcomplexes of aspherical 2-complexes are aspherical iff their fundamental groups have cohomological dimension 2.

Certain projective modules over group rings vanish under specified conditions for groups of cohomological dimension 2.

Abstract

A finite connected 2-complex K whose fundamental group is of cohomological dimension 2 is aspherical iff the subgroup \Sigma_K of H_2(K) consisting of spherical 2-cycles is zero. A finite connected subcomplex of an aspherical 2-complex is aspherical iff its fundamental group is of cohomological dimension 2. If G is a countable group such that extension of scalars from Z[G] to \ell_2(G) kills \bar K_0(Z[G]), and if P is a finitely generated projective Z[G]-module with P/IP=0, where I is the augmentation ideal of Z[G], then P=0. In particular, if G is a countable group of cohomological dimension 2 and P is a finitely generated projective Z[G]-module such that P/IP=0, then P=0.