Asphericity for certain groups of cohomological dimension 2

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A finite connected 2-complex K whose fundamental group is of cohomological dimension 2 is aspherical iff the subgroup Σₖ of H₂(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 ℓ₂(G) kills K₀(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.