What does this research mean for the field?

For all k ≥ 2, there exists a non-free stably free \( \mathbb{Z}G \)-module of rank k, leading to the existence of homotopically distinct finite 2-complexes with fundamental group G and Euler characteristic exceeding the minimal value over G by k. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This research aims to establish the existence of non-free stably free \( \mathbb{Z}G \)-modules and their implications for 2-complexes.

February 20, 2026Open Access

Stably free modules and the unstable classification of 2-complexes

Key Points

This research aims to establish the existence of non-free stably free \( \mathbb{Z}G \)-modules and their implications for 2-complexes.
Identification of a group G and construction of a non-free stably free \( \mathbb{Z}G \)-module of rank k.
Analysis of homotopically distinct finite 2-complexes with fundamental group G.
Examination of Euler characteristic values in relation to G.
Existence of non-free stably free \( \mathbb{Z}G \)-modules for all \( k \geq 2 \).
Discovery of homotopically distinct finite 2-complexes with Euler characteristics exceeding minimal values by k.
Resolution of Problem D5 from C. T. C. Wall's 1979 Problem List.

Abstract

Abstract 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.

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Stably free modules and the unstable classification of 2-complexes

Key Points

Abstract

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