#### Volume 8, issue 1 (2008)

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

### F Rudolf Beyl and Nancy Waller

Algebraic & Geometric Topology 8 (2008) 1–17
##### Abstract

This is a continuation of our study [A stably free nonfree module and its relevance for homotopy classification, case ${ℚ}_{28}$, Algebr Geom Topol 5 (2005) 899–910] of a family of projective modules over ${Q}_{4n}$, the generalized quaternion (binary dihedral) group of order $4n$. Our approach is constructive. Whenever $n\ge 7$ 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 $\theta :ℤG\oplus ℤG\cong P\oplus ℤG$ described by $2×2$ matrices. This makes the subject accessible to both theoretical and computational investigations, in particular, of Wall’s D(2)–problem.

##### Keywords
exotic algebraic 2-complex, Wall's D(2)-problem, stably free nonfree module, stabilizing isomorphism, homotopy classification of 2-complexes, truncated free resolution, generalized quaternion groups, single generation of modules, units in factor rings of integral group rings
##### Mathematical Subject Classification 2000
Primary: 16D40, 19A13, 57M20
Secondary: 55P15