3-manifold

In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe to a small enough observer. This is made more precise in the definition below.

Principles

Definition

A topological space is a 3-manifold if it is a second-countable Hausdorff space and if every point in has a neighbourhood that is homeomorphic to Euclidean 3-space.

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, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, and partial differential equations. 3-manifold theory is considered a part of low-dimensional topology or geometric topology.
A key idea in the theory is to study a 3-manifold by considering special surfaces 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 manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, 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. 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 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 Poincaré duality and the Hurewicz theorem, we have the following homology groups:

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 of.

Connected sums

One important topological operation is the connected sum 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 -module. For the special case of having each is infinite but not cyclic, if we take based embeddings of a 2-sphere

where

then the second fundamental group has the presentation

giving a straightforward computation of this group.

Important examples of 3-manifolds

Euclidean 3-space

Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it. This is just the standard 3-dimensional vector space over the real numbers.

3-sphere

A 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. Many examples of 3-manifolds can be constructed by taking quotients of the 3-sphere by a finite group acting freely on via a map, so.

Real projective 3-space

Real projective 3-space, or RP³, is the topological space of lines passing through the origin 0 in R⁴. It is a compact, smooth manifold of dimension 3, and is a special case Gr of a Grassmannian space.
RP³ is SO, hence admits a group structure; the covering map S³ → RP³ is a map of groups Spin → SO, where Spin is a Lie group that is the universal cover of SO.

3-torus

The 3-dimensional torus is the product of 3 circles. That is:
The 3-torus, T³ can be described as a quotient of R³ under integral shifts in any coordinate. That is, the 3-torus is R³ modulo the action of the integer lattice Z³. Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together.
A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group. Group multiplication on the torus is then defined by coordinate-wise multiplication.

Hyperbolic 3-space

Hyperbolic space is a homogeneous space that can be characterized by a constant negative curvature. It is the model of hyperbolic geometry. It is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and models of elliptic geometry that have a constant positive curvature. When embedded to a Euclidean space, every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the 3-ball in hyperbolic 3-space: it increases exponentially with respect to the radius of the ball, rather than polynomially.

Poincaré dodecahedral space

The Poincaré homology sphere is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. This shows the Poincaré conjecture cannot be stated in homology terms alone.
In 2003, lack of structure on the largest scales in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft.
However, there is no strong support for the correctness of the model, as yet.

Seifert–Weber space

In mathematics, Seifert–Weber space is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds.
It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space. Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space.
With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold.
It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.

Gieseking manifold

In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by.
The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner. Moreover, the angle made by the faces is. The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.