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

Abstract

This is a continuation of our study [3] of a family of projective modules over Q 4n , 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 ÂW ZG˚ZG Š 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. 16D40, 19A13, 57M20; 55P15

An exotic presentation of Q_28

“…For finite groups, presentations with distinct second homotopy but the same deficiency were found by Metzler for C 5 3 [25, 16, p.105]. The case of quaternion groups has been the subject of much analysis [14,15,3,4], largely because of its relation to Wall's D(2)-problem. In 1965 Wall showed that for n > 2, if a finite cell complex is cohomologically n dimensional (in the sense of having no non-trivial cohomology in dimensions above n with respect to any coefficient bundle), then it is in fact homotopy equivalent to an actual n dimensional cell complex [31].…”

“…Since the work of Johnson [14,15] and Beyl and Waller [3,4] in the early 2000's, the hunt has been on to find out if a finite balanced presentation of a quaternion group Q 4n can have non-standard second homotopy group. This has largely been fueled by the connection to Wall's famous D(2)-problem [15].…”

“…Based on this work, spaces were constructed which fell into the second category with fundamental group Q 2 k with k ≥ 5 [14] and fundamental group Q 28 [3,4]. That is, their second homotopy group was not IQ * 4n , and it was conjectured that no finite presentation of Q 4n would have a second homotopy group other than IQ * 4n .…”

Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction

“…where (1) follows from the definition of the action of AutpGq using ψ : AutpGq Ñ pZ{|G|q ˆ, and (2) follows from Theorem A.…”