A stably free nonfree module and its relevance for homotopy classification, caseQ₂₈

Abstract

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. AMS Classification 57M20; 55P15, 19A13Keywords 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 moduleThe main topic of this paper is the construction of an "exotic" algebraic 2-complex over Q 28 , the generalized quaternion group of order 28. The result provides significantly more detail than mere existence proofs, as the boundary maps are given by explicit matrices over the group ring (Section 4). This example should serve as a suitable test object in the investigation of an open problem of C.T.C. Wall [15, p.57], now [9] referred to as the D(2)-problem. D(2)-problem Suppose X is a finite three-dimensional connected CW-complex (with universal cover X ) such that H 3 ( X, Z) = 0 and H 3 (X, B) = 0 for all local coefficient systems B on X . Is X homotopy equivalent to a finite 2-complex?F.E.A. Johnson [7], [9] has shown that for finite base groups this question has an affirmative answer if, and only if, every algebraic 2-complex is geometrically realizable.The work of K.W. Gruenberg and P.A. Linnell [5, (2.6)] on minimal resolutions of lattices shows that stably equivalent lattices over ZQ 32 , despite stringent

On CW-complexes over groups with periodic cohomology

“…Our proof amounts to combining recent results of W. Mannan and T. Popiel [26] with Theorem 1.1. This group was proposed as a counterexample in [2].…”

“…Let i X : X (2) ֒→ X denote the inclusion and note that this induces a ZG chain map (i X ) * : C * (X (2) ) → C * (X) where the 2-skeleton X (2) comes equipped with the polarisation p X (2) = p X • π 1 (i X ), and similarly for Y (2) . Since (ϕ…”

On the $K$-theory and homotopy theory of the Klein bottle group

“…Let α n = (1)(y) n (y 3 ) n , β n = (1)(y −1 ) n (y −3 ) n (x) ∈ FF (x, y). It was shown in Section 2 that the elementsᾱ n = 1 + ny + ny 3 By Theorem 6.1 one would have to show that P = FG/ gα, gβ, g ∈ G = 1. We have not been able to adapt the arguments given for the trefoil group to the Klein bottle group.…”

“…Is X homotopy equivalent to a finite 2-complex? In [10, appendix B], Johnson proved the following realization theorem: Let G be a finitely presented group of type F L (3); then the D(2)-property holds for G if and only if every algebraic (G, 2)-complex admits a geometric realization. More information on the history of the geometric realization problem and the D(2)-problem can be found in the introduction of Johnson's book [10].…”

“…Other potential counterexamples to the geometric realization and the D(2) problem are in the literature. See, for instance, Beyl-Waller [3]. A finitely gener-…”