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

Beyl, F Rudolf¹ ; Waller, Nancy¹

¹Department of Mathematics and Statistics, Portland State University, Portland, OR 97207-0751, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 1-17

Résumé

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.

Détail

DOI : 10.2140/agt.2008.8.1

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

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