Stably free modules and the unstable classification of 2-complexes

TL;DR

This paper demonstrates the existence of non-free stably free modules over group rings and uses this to classify 2-complexes with the same fundamental group but different topological properties, resolving a longstanding problem.

Contribution

It proves the existence of non-free stably free modules over group rings for all ranks ≥ 2 and applies this to classify 2-complexes with the same fundamental group but different Euler characteristics.

Findings

Existence of non-free stably free modules over  G for all ranks  

Construction of homotopically distinct 2-complexes with same  but different Euler characteristics

Resolution of Problem D5 in C. T. C. Wall's 1979 Problem List

Abstract

For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, 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.