Stably free modules and the unstable classification of 2-complexes

TL;DR

The paper resolves Wall’s D5 by constructing, for every $k\ge2$, a group $G$ and a non-free stably free $\mathbb{Z}G$-module of rank $k$, showing non-cancellation phenomena in unstable classifications of projective modules and finite $2$-complexes. It proves that with $G=\ast_{i=1}^k T$ (trefoil groups) one obtains infinitely many distinct stably free $\mathbb{Z}G$-modules of rank $k$, with corresponding $(G,2)$-complexes $X$ and $Y$ that are homotopy-distinct yet satisfy $X\vee S^2\simeq Y\vee S^2$. The work extends to higher cohomological dimensions via iterated group constructions $G_{(n)}$ and provides a framework tying stable, algebraic data to unstable geometric classifications, including potential implications for open problems in smooth $4$-manifold topology. It also develops a robust theory of induced module decompositions over free products, linking Bergman-type results to topological realizations and highlighting both cancellation phenomena and their limitations in the infinite-group setting.

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.

Figures (2)

• Figure 1: Branching phenomena for the graded trees $\mathcal{S}_c$. The vertical height is $\ell(\cdot) \in \mathbb{Z}_{\ge 0}$.
• Figure 2: Further branching phenomena

Theorems & Definitions (111)

• Theorem A
• Theorem B
• Theorem A$'$
• Theorem C
• Remark A$'$
• Theorem B$'$
• Theorem B$'$
• Theorem B$'$
• Remark B$'$
• Proposition B$'$