Wall’s D2 Problem and Stably-Free Modules over Integral Group Rings

Seminar/Forum

Wall’s D2 Problem and Stably-Free Modules over Integral Group Rings

In the 1960s, Wall asked for conditions for a finite CW-complex \(X\) to be homotopic to a complex of dimension \(n\). For \(n \ne 2\), a necessary and sufficient condition is that \[Hk(\widetilde{X};\mathbb{Z})=H^k(X; \mathcal{B})=0\] for all \(k > n\) and coefficient bundles \(\mathcal{B}\). The D2 problem asks whether or not these conditions are still sufficient in the case \(n=2\). Results of Johnson, from the early 2000s, cemented a link between the D2 problem for complexes with \(\pi1(X)=G\) and stable modules over \(\mathbb{Z}[G]\). This led to an affirmative solution to the D2 problem if \(\pi_1(X)\) was one of a large class of groups \(G\), with a key result requiring that \(\mathbb{Z}[G]\) has stably free cancellation (SFC), i.e. no non-trivial stably-free modules. More recently, Beyl and Waller showed that non-trivial stably free modules over \(\mathbb{Z}[G]\), for certain groups \(G\), can be used to construct 3-complexes which are potential counterexamples for the D2 problem. I will discuss some recent progress made on the problem of classifying all finite groups \(G\) for which \(\mathbb{Z}[G]\) has stably free cancellation (SFC). In particular, we extend results of R. G. Swan by giving a condition for SFC and use this show that \(\mathbb{Z}[G]\) has SFC provided at most one copy of the quaternions occurs in the Wedderburn decomposition of \(\mathbb{R}[G]\). This gives a generalisation of the Eichler condition in the case of integral group rings, and places large restrictions on the possible fundamental groups of “exotic” 3-complexes which can be constructed using methods similar to the ones used above.

Presenter

  •  Johnny Nicholson
    Johnny Nicholson, University College London