Stably free modules and the unstable classification of 2-complexes

For all k ≥ 2 k ≥ 2 , we show that there exists a group G and a non-free stably free ZG Z G -module of rank k . We use this to show that, for all k ≥ 2 k ≥ 2 , there exist homotopically distinct finite 2-complexes with fundamental group G and with Euler characteristic exceeding the minimal value over G by k . This resolves Problem D5 in the 1979 Problem List of C. T. C. Wall. We also explore a number of generalisations and present a potential application to the topology of closed smooth 4-manifolds.