In
mathematics, a 3-manifold is a
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible
shape of the universe. Just as a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
looks like a
plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
Introduction
Definition
A
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is a 3-manifold if it is a
second-countable Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
and if every point in ''X'' has a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
that is
homeomorphic to
Euclidean 3-space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
.
Mathematical theory of 3-manifolds
The topological,
piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.
Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. This special role has led to the discovery of close connections to a diversity of other fields, such as
knot theory,
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
,
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
,
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
,
Teichmüller theory,
topological quantum field theory,
gauge theory,
Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer in ...
, and
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
. 3-manifold theory is considered a part of
low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot th ...
or
geometric topology.
A key idea in the theory is to study a 3-manifold by considering special
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an
incompressible surface and the theory of
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in w ...
s, or one can choose the complementary pieces to be as nice as possible, leading to structures such as
Heegaard splitting
In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Definitions
Let ''V'' and ''W'' be handlebodies of genus ''g'', an ...
s, which are useful even in the non-Haken case.
Thurston's contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry (of which there are eight). The most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful.
The
fundamental groups of 3-manifolds strongly reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
and topological methods.
Invariants describing 3-manifolds
3-manifolds are an interesting special case of low-dimensional topology because their topological invariants give a lot of information about their structure in general. If we let
be a 3-manifold and
be its fundamental group, then a lot of information can be derived from them. For example, using
Poincare duality and the
Hurewicz theorem, we have the following
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s:
where the last two groups are isomorphic to the
group homology and cohomology of
, respectively; that is,
From this information a basic homotopy theoretic classification of 3-manifolds can be found. Note from the
Postnikov tower there is a canonical map
If we take the pushforward of the fundamental class
into
we get an element
. It turns out the group
together with the group homology class
gives a complete algebraic description of the
homotopy type
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
of
.
Connected sums
One important topological operation is the
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
of two 3-manifolds
. In fact, from general theorems in topology, we find for a three manifold with a connected sum decomposition
the invariants above for
can be computed from the
. In particular
Moreover, a 3-manifold
which cannot be described as a connected sum of two 3-manifolds is called prime.
Second homotopy groups
For the case of a 3-manifold given by a connected sum of prime 3-manifolds, it turns out there is a nice description of the second fundamental group as a