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Poincaré duality in dimension 3

C T C Wall

Geometry & Topology Monographs 7 (2004) 1–26

DOI: 10.2140/gtm.2004.7.1

Abstract

The paper gives a review of progress towards extending the Thurston programme to the Poincaré duality case. In the first section, we fix notation and terminology for Poincaré complexes X (with fundamental group G) and pairs, and discuss finiteness conditions.

For the case where there is no boundary, π2 is non-zero if and only if G has at least 2 ends: here one would expect X to split as a connected sum. In fact, Crisp has shown that either G is a free product, in which case Turaev has shown that X indeed splits, or G is virtually free. However very recently Hillman has constructed a Poincaré complex with fundamental group the free product of two dihedral groups of order 6, amalgamated along a subgroup of order 2.

In general it is convenient to separate the problem of making the boundary incompressible from that of splitting boundary-incompressible complexes. In the case of manifolds, cutting along a properly embedded disc is equivalent to attaching a handle along its boundary and then splitting along a 2–sphere. Thus if an analogue of the Loop Theorem is known (which at present seems to be the case only if either G is torsion-free or the boundary is already incompressible) we can attach handles to make the boundary incompressible. A very recent result of Bleile extends Turaev's arguments to the boundary-incompressible case, and leads to the result that if also G is a free product, X splits as a connected sum.

The case of irreducible objects with incompressible boundary can be formulated in purely group theoretic terms; here we can use the recently established JSJ type decompositions. In the case of empty boundary the conclusion in the Poincaré duality case is closely analogous to that for manifolds; there seems no reason to expect that the general case will be significantly different.

Finally we discuss geometrising the pieces. Satisfactory results follow from the JSJ theorems except in the atoroidal, acylindrical case, where there are a number of interesting papers but the results are still far from conclusive.

The latter two sections are adapted from the final chapter of my survey article on group splittings.

Keywords

Poincaré complex, splitting, loop theorem, incompressible, JSJ theorem, geometrisation

Mathematical Subject Classification

Primary: 57P10

References
Publication

Received: 22 October 2003
Revised: 29 March 2004
Accepted: 10 March 2004
Published: 17 September 2004

Authors
C T C Wall
Department of Mathematical Sciences
University of Liverpool
Liverpool
L69 3BX
UK