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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


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.

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
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Received: 10 February 2005
Accepted: 1 June 2005
Published: 29 July 2005
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