Stably free modules and the unstable classification of 2-complexes

2021/08/04 by Nicholson, John · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2108.02220

Abstract

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