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Examples of exotic free 2–complexes and stably free
nonfree modules for quaternion groups
F Rudolf Beyl and Nancy Waller
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Algebraic & Geometric Topology 8
(2008) 1–17
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Abstract
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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.
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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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Mathematical Subject Classification
Primary: 16D40, 19A13, 57M20
Secondary: 55P15
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Publication
Received: 5 July 2007
Accepted: 5 September 2007
Published: 8 February 2008
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