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Department of Mathematical Sciences

# Seminar Archives

## Pure Maths Colloquium: Indecomposable $PD_3$-complexes and virtually free groups

Presented by Jonathan Hillman, University of Sydney (Australia)

20 October 2008 16:15 in CM221

Poincar\'e duality complexes model the algebraic topology of manifolds. It is known that in dimensions 1 and 2 all such complexes are homotopy equivalent to manifolds. In his foundational 1967 work on such complexes Wall raised several questions about the 3-dimensional case, motivated by knowledge of 3-manifolds. In the 1980s Turaev answered one of these questions when he showed that such a complex decomposes as a sum if and only if its fundamental group is a free product. Experience with 3-manifolds suggested that indecomposable (orientable) $PD_3$-complexes should either be aspherical, $S^1\times{S^2}$, or have finite fundamental group. We shall describe a counterexample to this expectation, with fundamental group virtually free, and outline some recent work suggesting that any other counterexample must have a similar structure. [The example has just 8 cells, and its construction is quite explicit. The (very) recent work involves using repeatedly a result of Crisp, that centralizers of elements of odd order in the fundamental group must be finite, applied to graphs of finite groups.]