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

## Seminar/Forum

107
Peter Hall

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.