Volume 6, issue 5 (2006)

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Exotic relation modules and homotopy types for certain 1–relator groups

Jens Harlander and Jacqueline A Jensen

Algebraic & Geometric Topology 6 (2006) 2163–2173
 arXiv: 0906.5491
Abstract

Using stably free non-free relation modules we construct an infinite collection of 2–dimensional homotopy types, each of Euler-characteristic one and with trefoil fundamental group. This provides an affirmative answer to a question asked by Berridge and Dunwoody [J. London Math. Soc. 19 (1979) 433–436]. We also give new examples of exotic relation modules. We show that the relation module associated with the generating set $\left\{x,{y}^{4}\right\}$ for the Baumslag–Solitar group $〈x,y\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}x{y}^{2}{x}^{-1}={y}^{3}〉$ is stably free non-free of rank one.

Keywords
2-dimensional complex, homotopy-type, stably free modules
Primary: 57M20
Secondary: 57M05