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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

This is a continuation of our study [A stably free nonfree module and its relevance for homotopy classification, case 28, 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: G GP G described by 2×2 matrices. This makes the subject accessible to both theoretical and computational investigations, in particular, of Wall’s D(2)–problem.

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
Received: 5 July 2007
Accepted: 5 September 2007
Published: 8 February 2008
F Rudolf Beyl
Department of Mathematics and Statistics
Portland State University
Portland, OR 97207-0751
Nancy Waller
Department of Mathematics and Statistics
Portland State University
Portland, OR 97207-0751