## Algebraic & Geometric Topology

### Examples of exotic free $2$–complexes and stably free nonfree modules for quaternion groups

#### Abstract

This is a continuation of our study [A stably free nonfree module and its relevance for homotopy classification, case $ℚ28$, Algebr Geom Topol 5 (2005) 899–910] of a family of projective modules over $Q4n$, the generalized quaternion (binary dihedral) group of order $4n$. Our approach is constructive. Whenever $n≥7$ is odd, this work provides examples of stably free nonfree modules of rank $1$, which are then used to construct exotic algebraic $2$–complexes relevant to Wall’s D(2)–problem. While there are examples of stably free nonfree modules for many infinite groups $G$, there are few actual examples for finite groups. This paper offers an infinite collection of finite groups with stably free nonfree modules $P$, given as ideals in the group ring. We present a method for constructing explicit stabilizing isomorphisms $θ:ℤG⊕ℤG≅P⊕ℤG$ described by $2×2$ matrices. This makes the subject accessible to both theoretical and computational investigations, in particular, of Wall’s D(2)–problem.

#### Article information

Source
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 1-17.

Dates
Accepted: 5 September 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796805

Digital Object Identifier
doi:10.2140/agt.2008.8.1

Mathematical Reviews number (MathSciNet)
MR2377275

Zentralblatt MATH identifier
1173.57002

#### Citation

Beyl, F Rudolf; Waller, Nancy. Examples of exotic free $2$–complexes and stably free nonfree modules for quaternion groups. Algebr. Geom. Topol. 8 (2008), no. 1, 1--17. doi:10.2140/agt.2008.8.1. https://projecteuclid.org/euclid.agt/1513796805

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