3-dimensional Topology

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 (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

Principles

Definition

A topological space $M$ is a 3-manifold if it is a second-countable Hausdorff space and if every point in $M$ 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 (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 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 $M$ be a 3-manifold and $\pi = \pi_1(M)$ 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:

\begin
H_0(M) &= H^3(M) =& \mathbb \\
H_1(M) &= H^2(M) =& \pi/ pi,\pi\\
H_2(M) &= H^1(M) =& \text(\pi,\mathbb) \\
H_3(M) &= H^0(M) = & \mathbb
\end

where the last two groups are isomorphic to the group homology and cohomology of $\pi$, respectively; that is,

\begin
H_1(\pi;\mathbb) &\cong \pi/ pi,\pi\\
H^1(\pi;\mathbb) &\cong \text(\pi,\mathbb)
\end

From this information a basic homotopy theoretic classification of 3-manifolds can be found. Note from the Postnikov tower there is a canonical map

$q: M \to B\pi$

If we take the pushforward of the fundamental class \in H_3(M) into $H_3(B\pi)$ we get an element \zeta_M = q_*( . It turns out the group $\pi$ together with the group homology class $\zeta_M \in H_3(\pi,\mathbb)$ gives a complete algebraic description of the homotopy type of $M$.

Connected sums

One important topological operation is the connected sum of two 3-manifolds $M_1\# M_2$. In fact, from general theorems in topology, we find for a three manifold with a connected sum decomposition $M = M_1\# \cdots \# M_n$ the invariants above for $M$ can be computed from the $M_i$. In particular

$\begin H_1(M) &= H_1(M_1)\oplus \cdots \oplus H_1(M_n) \\ H_2(M) &= H_2(M_1)\oplus \cdots \oplus H_2(M_n) \\ \pi_1(M) &= \pi_1(M_1) * \cdots * \pi_1(M_n) \end$

Moreover, a 3-manifold $M$ 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 \mathbb pi/math>-module. For the special case of having each $\pi_1(M_i)$ is infinite but not cyclic, if we take based embeddings of a 2-sphere

$\sigma_i:S^2 \to M$ where $\sigma_i(S^2) \subset M_i - \ \subset M$

then the second fundamental group has the presentation

$\pi_2(M) = \frac$

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 (or 2-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 $\pi$ acting freely on $S^3$ via a map $\pi \to \text(4)$, so $M = S^3/\pi$.

Real projective 3-space

Real projective 3-space, or RP^''3'', 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(1, R⁴) of a Grassmannian space.

RP³ is (diffeomorphic to) SO(3), hence admits a group structure; the covering map ''S''³ → RP³ is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3).

3-torus

The 3-dimensional torus is the product of 3 circles. That is:

:$\mathbf^3 = S^1 \times S^1 \times S^1.$

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 (with the action being taken as vector addition). 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 (when identified with the unit complex numbers with multiplication). 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 (like the 3-sphere) that have a constant positive curvature. When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the 3-ball

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