Algebraic & Geometric Topology

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

F Rudolf Beyl and Nancy Waller

Full-text: Open access

Abstract

This is a continuation of our study [A stably free nonfree module and its relevance for homotopy classification, case 28, 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 n7 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:GGPG described by 2×2 matrices. This makes the subject accessible to both theoretical and computational investigations, in particular, of Wall’s D(2)–problem.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 1-17.

Dates
Received: 5 July 2007
Accepted: 5 September 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796805

Digital Object Identifier
doi:10.2140/agt.2008.8.1

Mathematical Reviews number (MathSciNet)
MR2377275

Zentralblatt MATH identifier
1173.57002

Subjects
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

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

Citation

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


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