Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 8, Number 1 (2008), 1-17.
Examples of exotic free $2$–complexes and stably free nonfree modules for quaternion groups
This is a continuation of our study [A stably free nonfree module and its relevance for homotopy classification, case , Algebr Geom Topol 5 (2005) 899–910] of a family of projective modules over , the generalized quaternion (binary dihedral) group of order . Our approach is constructive. Whenever is odd, this work provides examples of stably free nonfree modules of rank , which are then used to construct exotic algebraic –complexes relevant to Wall’s D(2)–problem. While there are examples of stably free nonfree modules for many infinite groups , there are few actual examples for finite groups. This paper offers an infinite collection of finite groups with stably free nonfree modules , given as ideals in the group ring. We present a method for constructing explicit stabilizing isomorphisms described by matrices. This makes the subject accessible to both theoretical and computational investigations, in particular, of Wall’s D(2)–problem.
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 1-17.
Received: 5 July 2007
Accepted: 5 September 2007
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 16D40: Free, projective, and flat modules and ideals [See also 19A13] 19A13: Stability for projective modules [See also 13C10] 57M20: Two-dimensional complexes
Secondary: 55P15: Classification of homotopy type
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
Beyl, F Rudolf; Waller, Nancy. Examples of exotic free $2$–complexes and stably free nonfree modules for quaternion groups. Algebr. Geom. Topol. 8 (2008), no. 1, 1--17. doi:10.2140/agt.2008.8.1. https://projecteuclid.org/euclid.agt/1513796805