SMS scnews item created by Boris Lishak at Mon 20 Aug 2018 1237
Type: Seminar
Modified: Wed 22 Aug 2018 1017
Distribution: World
Calendar1: 22 Aug 2018 1200-1300
CalLoc1: Carslaw 830
CalTitle1: Nicholson -- Poincare 3-complexes and Stably-Free Modules over Integral Group Rings
Auth: borisl@dora.maths.usyd.edu.au

# Poincare 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].

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