Poincare 3-complexes and Stably-Free Modules over Integral Group Rings

Abstract:

The development of surgery theory during the 1960s led Wall to ask for general homotopical conditions which must be satisfied by a space before it can be transformed, by surgery, into a manifold. Notably the space must be a Poincaré \(n\)-complex but, as is the case for manifolds, it must also admit a finite cell structure with a single n-cell. This is known to be true for all Poincaré \(n\)-complexes except in the case \(n=3\) which remains open. This is equivalent to a special case of Wall’s D2 problem which asks for conditions for a finite CW-complex X to be homotopic to a complex of dimension \(n\).

Results of Johnson, from the early 2000s, cemented a link between the D2 problem for complexes with \( \pi_1(X)=G \) 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 \(\mathbb{H}\) 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.

See arXiv:1807.00307 [math.KT].