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].