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 ${\mathbb{Q}}_{28}$, 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 $\theta :\mathbb{Z}G\oplus \mathbb{Z}G\cong P\oplus \mathbb{Z}G$ described by $2\times 2$ matrices. This makes the subject accessible to both theoretical and computational investigations, in particular, of Wall’s D(2)–problem.