The homotopy type of a finite 2-complex with non-minimal Euler characteristic

Abstract

We resolve two long-standing and closely related problems concerning stably free ZG-modules and the homotopy type of finite 2-complexes. In particular, for all k ≥ 1, we show that there exists a group G and a non-free stably free ZG-module of rank k. We use this to show that, for all k ≥ 0, there exists homotopically distinct finite 2-complexes with fundamental group G and with Euler characteristic k greater than the minimal value over G. This provides a solution to Problem D5 in the 1979 Problems List of C. T. C. Wall.