Geometry & Topology Monographs, Vol. 7 (2004),
Proceedings of the Casson Fest,
Paper no. 1, pages 1--26.
Poincare duality in dimension 3
CTC Wall
Abstract.
The paper gives a review of progress towards extending the Thurston
programme to the Poincare duality case. In the first section, we fix
notation and terminology for Poincare complexes X (with fundamental
group G) and pairs, and discuss finiteness conditions.
For the case
where there is no boundary, pi_2 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 Poincare 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 Poincare 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.
Poincare complex, splitting, loop theorem, incompressible, JSJ theorem, geometrisation
AMS subject classification.
Primary: 57P10.
Secondary: \thesecondaryclass .
E-print: arXiv:math.GT/0410043
Submitted to GT on 22 October 2003.
(Revised 29 March 2004.)
Paper accepted 10 March 2004.
Paper published 17 September 2004.
Notes on file formats
CTC Wall
Department of Mathematical Sciences, University of Liverpool
Liverpool, L69 3BX, UK
Email: ctcw@liv.ac.uk
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