Algebraic & Geometric Topology

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

Jens Harlander and Jacqueline A Jensen

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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 {x,y4} for the Baumslag–Solitar group x,y|xy2x1=y3 is stably free non-free of rank one.

Article information

Algebr. Geom. Topol., Volume 6, Number 5 (2006), 2163-2173.

Received: 18 May 2006
Accepted: 25 September 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M20: Two-dimensional complexes
Secondary: 57M05: Fundamental group, presentations, free differential calculus

2-dimensional complex homotopy-type stably free modules


Harlander, Jens; Jensen, Jacqueline A. Exotic relation modules and homotopy types for certain 1–relator groups. Algebr. Geom. Topol. 6 (2006), no. 5, 2163--2173. doi:10.2140/agt.2006.6.2163.

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