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<p>For all <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that there exists a group <i>G</i> and a non-free stably free <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {Z}G\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>-module of rank <i>k</i>. We use this to show that, for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(k \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, there exist homotopically distinct finite 2-complexes with fundamental group <i>G</i> and with Euler characteristic exceeding the minimal value over <i>G</i> by <i>k</i>. 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.</p>

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Stably free modules and the unstable classification of 2-complexes

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月18日

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  • John Nicholson

摘要

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