Poincaré duality is central to the understanding of manifold topology.
Dimension 3 is critical in various respects, being between the known
territory of surfaces and the wilderness manifest in dimensions
$\ge $ 4.
The main thrust of 3manifold topology for the past half century has been to show
that aspherical closed 3manifolds are determined by their fundamental groups.
Relatively little attention has been given to the question of
which groups
arise. This book is the first comprehensive account of what is known about
PD${}_{3}$complexes,
which model the homotopy types of closed 3manifolds, and
PD${}_{3}$groups,
which correspond to aspherical 3manifolds. In the first half we show that every
P${}^{2}$irreducible
PD${}_{3}$complex
is a connected sum of indecomposables, which are either aspherical or
have virtually free fundamental group, and largely determine the latter
class. The picture is much less complete in the aspherical case. We sketch
several possible aproaches for tackling the central question, whether every
PD${}_{3}$group
is a 3manifold group, and then explore properties of subgroups of
PD${}_{3}$groups,
unifying many results of 3manifold topology. We conclude with an appendix listing
over 60 questions. Our general approach is to prove most assertions which are
specifically about Poincaré duality in dimension 3, but otherwise to cite standard
references for the major supporting results.
Target readership: graduate students and mathematicians with an interest in
lowdimensional topology.
