#### Volume 5, issue 3 (2005)

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A stably free nonfree module and its relevance for homotopy classification, case $Q_{28}$

### F Rudolf Beyl and Nancy Waller

Algebraic & Geometric Topology 5 (2005) 899–910
 arXiv: math.RA/0508196
##### Abstract

The paper constructs an “exotic” algebraic 2–complex over the generalized quaternion group of order 28, with the boundary maps given by explicit matrices over the group ring. This result depends on showing that a certain ideal of the group ring is stably free but not free. As it is not known whether the complex constructed here is geometrically realizable, this example is proposed as a suitable test object in the investigation of an open problem of C T C Wall, now referred to as the D(2)–problem.

##### Keywords
algebraic 2–complex, Wall's D(2)–problem, geometric realization of algebraic 2–complexes, homotopy classification of 2–complexes, generalized quaternion groups, partial projective resolution, stably free nonfree module
##### Mathematical Subject Classification 2000
Primary: 57M20
Secondary: 55P15, 19A13