Stable modules and Wall's D(2)-problem

Abstract

Abstract. The D(2)-problem is to determine whether for a three-dimensional complex X, the vanishing of 3-dimensional cohomology, in all coefficients, is enough to guarantee that X is homotopically two-dimensional. We show that for finite complexes with finite fundamental group, a positive solution to the D(2)-problem is obtained precisely when all stably free algebraic 2-complexes are geometrically realizable.The proof makes very strong use of techniques which apply to finite fundamental groups but not more generally; in particular, Yoneda's Theorem that, for finite groups, group cohomology is representable by stable modules of finite type, and also the Swan-Jacobinski Cancellation Theorem for such stable modules.

Explicit homotopy equivalences in dimension two

“…It is a fundamental problem, first encountered by Wall in his study of geometric homotopy theory [19], to decide whether a finitely presented group G possesses the D(2)-property, defined by requiring that every finite 3-complex X G with π 1 (X G ) % G which is cohomologically two dimensional should be homotopy equivalent to a finite 2-complex. In [10] we proved:…”

Minimal 2-complexes and the D(2)-problem

“…It follows from the main result of [6], however, that if χ * (n) is not realisable, then the D(2)-property fails for Q(2 n ); that is:…”

“…In this paper we consider only the case where G is finite; then there is a well-defined stable module Ω 3 (Z) which contains all such possible algebraic homotopy groups J. E is minimal when rk Z (J) attains the minimum possible value within Ω 3 (Z). Browning's Theorem ( [1]) essentially shows that the Realization Problem is equivalent to the problem of realizing minimal 2-complexes ( [1], [4]; see also [6]). Let Q(4m) denote the quaternion group of order 4m, given in the following standard presentation:…”

“…In [6], we showed that, for finite fundamental groups G, the D(2)-problem is equivalent to the Realization Problem. The question of whether X * (n) is realizable up to homotopy as the Cayley complex of some finite presentation is an intriguing one.…”

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

“…This is equivalent to showing that every chain homotopy type of algebraic 2-complexes is geometrically realizable, ie, the homotopy class contains a chain complex of the universal cover of a presentation complex for G [9]. Johnson [8,Theorem IV,p 220] showed that the chain homotopy classes of algebraic 2-complexes of minimal Euler characteristic over Q 4n correspond to the isomorphism classes of rank 1 stably free ZQ 4n -modules.…”