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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.
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