Wall’s D2 Problem and StablyFree Modules over Integral Group Rings
Seminar/Forum
In the 1960s, Wall asked for conditions for a finite CWcomplex \(X\) 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 nontrivial stablyfree modules. More recently, Beyl and Waller showed that nontrivial stably free modules over \(\mathbb{Z}[G]\), for certain groups \(G\), can be used to construct 3complexes 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” 3complexes which can be constructed using methods similar to the ones used above.
Presenter

Johnny Nicholson, University College London