Homotopy classes of truncated projective resolutions (Q1312813)

If \(X\) is a finite connected \(m\)-dimensional CW-complex whose universal cover \(\widetilde{X}\) is \((m-1)\)-connected, then the homotopy type of \(X\) can be recognized from the homotopy type of the cellular chain complex \(C(\widetilde{X})\). This result, due to \textit{S. MacLane} and \textit{J. H. C. Whitehead} [Proc. Natl. Acad. Sci. USA 36, 41-48 (1950; Zbl 0035.390)], transforms a topological problem into a question about the integral representation of the fundamental groups. The case, in which these groups are assumed to be finite, was studied by \textit{W. Browning} [unpublished (1970)]: the chain homotopy classes of truncated projective \(\mathbb{Z} G\)-resolutions of \(\mathbb{Z}\) for a given finite group \(G\) can be parametrized by the elements of a certain naturally occurring group within the \(K\)-theory of \(G\). The present paper gives a new treatment of the main results using only concepts from elementary \(K\)-theory (Browning works with pointed lattices).