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Algebra Seminar
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Jonathan Hillman
University of Sydney

Decomposition of PD3-complexes

Friday 15th October, 12:05-12:55pm, Carslaw 175.

Every closed orientable 3-manifold has an essentially unique factorization as a connected sum of prime 3-manifolds, which either have finite fundamental group or are aspherical or are homeomorphic to S1 × S2. As PDn-complexes model the homotopy type of n-manifolds it is natural to ask whether a similar result holds for PD3-complexes. Turaev showed that a PD3-complex P is a connected sum if and only if its fundamental group is a free product, and Crisp showed that if P is indecomposable then it is aspherical or has virtually free fundamental group. We shall review briefly their results, and give an example of an indecomposable PD3-complex whose fundamental group is virtually free but not virtually cyclic. This represents a departure from what 3-manifold theory would suggest, and is a counter-example to a question raised by C.T.C.Wall in 1967.