A sutured manifold is a tool in geometric topology which was first introduced by David Gabai in order to study taut
foliations on 3-manifolds. Roughly, a sutured manifold
is a pair 
with 
a compact, oriented
3-manifold with boundary and with 
a set of simple closed
curves in 
which are oriented and which divide 
into pieces 
and 
(Juhász 2010).
Defined precisely in a seminal work by Gabai (1983), a sutured manifold 
is a compact oriented 3-manifold 
together with a set 
of pairwise disjoint annuli 
and tori 
such that each component of 
contains a homologically
nontrivial oriented simple closed curve (called a suture) and such that 
is oriented. Using this construction,
the collection 
of a sutured manifold 
effectively splits 
into disjoint pieces 
and 
with 
, respectively 
, defined to be the components of 
whose normal
vectors point into, respectively point out of, 
. Gabai's definition also requires that orientations on 
be coherent with respect to the
set 
of sutures in the sense that any component 
of 
with boundary orientation must represent the
same homology class in 
as some suture.
The study of sutured 3-manifolds has yielded several strong results and continues to be an important focus of research among topologists today. For example, Gabai's
work on sutured 3-manifolds provided the framework necessary to obtain answers to
several longstanding problems including the Poenaru conjecture and the Property R
conjecture, as well as a number of knot-theoretic
problems including the superadditivity of knot genus and property
P for satellite knots (Scharlemann 1989). In
addition, sutured 3-manifolds which are balanced (i.e., those manifolds 
which have no closed components, for which each component
of 
contains a suture, and for which 
where 
denotes the Euler characteristic)
have also been studied in the context of so-called sutured Floer homology, an invariant
of balanced sutured manifolds and a generalization of both Heegaard Floer homology
and knot Floer homology (Juhász 2010).
