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

John Nicholson (UCL)

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].