3-manifold

In mathematics, a 3-manifold is a special kind of space that looks like our everyday three-dimensional world when you zoom in close enough. Imagine standing on a sphere—it looks flat to you because you're so close to its surface. The same idea applies to a 3-manifold: no matter where you look, it seems like normal space, just like our universe.

A 3-manifold can be thought of as one of the possible shapes the universe might have. Scientists and mathematicians study these shapes to better understand the structure of space and the universe around us.

This idea helps us explore complex shapes and spaces in a way that connects to our own experience of the world.

Principles

A 3-manifold is a special kind of space in mathematics. Imagine our world as a three-dimensional space — up, down, left, right, forward, and backward. A 3-manifold is like a shape that looks the same as our world if you look at it from very close up. No matter how you twist or bend it, to a tiny observer, it always seems like normal space.

Mathematicians study 3-manifolds to understand the possible shapes of the whole universe. These shapes can be very different from each other, and they connect to many other areas of math, like the study of knots or shapes with special symmetry. By looking at slices or cross-sections of these shapes, experts can learn a lot about their overall structure.

Important examples of 3-manifolds

Euclidean 3-space

Main article: Euclidean 3-space

Euclidean 3-space is a very important example of a 3-manifold. It is the standard three-dimensional space that we all live in, and other 3-manifolds are often described in relation to it.

3-sphere

Main article: 3-sphere

A 3-sphere is like a three-dimensional version of a regular sphere. While a regular sphere is the surface of a ball in three dimensions, a 3-sphere is the surface of a ball in four dimensions.

Real projective 3-space

Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect have infinite radius (= straight line).

Main article: Real projective space

Real projective 3-space, written as RP³, is the space of all straight lines that pass through a central point in four-dimensional space. It has three dimensions and can be thought of as a special kind of three-dimensional space.

3-torus

Main article: Torus § n-dimensional torus

The 3-torus is formed by taking three circles and combining them together. It can also be made by taking a three-dimensional cube and gluing the opposite faces together.

Hyperbolic 3-space

Main article: hyperbolic 3-space

Hyperbolic 3-space is a special kind of three-dimensional space that has constant negative curvature. This means that it looks different from regular Euclidean space in important ways. For example, the space covered by a ball grows much faster in hyperbolic space than in regular space.

A perspective projection of a dodecahedral tessellation in H3.Four dodecahedra meet at each edge, and eight meet at each vertex, like the cubes of a cubic tessellation in E3

Poincaré dodecahedral space

Main article: Homology sphere § Poincaré homology sphere

The Poincaré dodecahedral space is a special kind of three-dimensional space that can be made by gluing the faces of a dodecahedron (a twelve-sided shape) together in a particular way.

Seifert–Weber space

Main article: Seifert–Weber space

The Seifert–Weber space is another special kind of hyperbolic three-dimensional space. It can be made by gluing the faces of a dodecahedron together with a specific turning pattern.

Gieseking manifold

Main article: Gieseking manifold

The Gieseking manifold is a special kind of three-dimensional space with certain properties. It can be constructed by taking a four-sided shape (a tetrahedron) and gluing its faces together in pairs.

Some important classes of 3-manifolds

Hyperbolic link complements

A hyperbolic link is a special kind of shape in a three-dimensional space that has a special kind of measurement called a Riemannian metric with constant negative curvature. This means it follows a type of geometry called hyperbolic geometry. A hyperbolic knot is a hyperbolic link that has just one part.

Some well-known examples include:

These groups can sometimes overlap, meaning one example might fit into more than one group.

Some important structures on 3-manifolds

Contact geometry

Contact geometry is a way to study shapes in space. It looks at special rules that help us understand how things move and change. This idea is like another math idea called symplectic geometry, but it works in spaces with an odd number of dimensions.

Haken manifold

A Haken manifold is a special kind of 3D shape that can be split into simpler pieces. These shapes were first studied by a mathematician named Wolfgang Haken. He found ways to tell if a shape is a Haken manifold and how to break it apart.

Essential lamination

An essential lamination is a way to cover a 3D shape with layers. Each layer fits tightly and cannot be pushed away from the shape. This idea helps us understand more about Haken manifolds.

Heegaard splitting

A Heegaard splitting is a method to split a 3D shape into two simpler parts called handlebodies. Every closed 3D shape can be split this way, which is different from shapes in higher dimensions.

Taut foliation

A taut foliation is a way to cover a 3D shape with layers that all fit together in a special way. This idea was made famous by mathematicians William Thurston and David Gabai.

Main article: Contact geometry

Main article: Haken manifold

Main article: Heegaard splitting

Main article: Taut foliation

Foundational results

Some important ideas in this area are named as guesses because of their history.

We start with ideas about shapes:

Moise's theorem

Moise's theorem, proved by Edwin E. Moise, says that any three-dimensional space made of simple pieces has a special smooth shape.

Prime decomposition theorem

The prime decomposition theorem says that every three-dimensional space can be built from simpler pieces that cannot be broken down further.

Loop and Sphere theorems

The loop theorem and the sphere theorem give conditions for certain shapes to exist inside three-dimensional spaces.

Annulus and Torus theorems

The annulus theorem and the torus theorem describe when certain ring-like or doughnut-like shapes can be found inside three-dimensional spaces.

JSJ decomposition

The JSJ decomposition is a way to cut a three-dimensional space along special surfaces so that each piece has a simple shape.

Scott core theorem

The Scott core theorem says that for any three-dimensional space, there is a smaller space inside it that keeps all the important information.

Lickorish–Wallace theorem

The Lickorish–Wallace theorem says that any three-dimensional space can be made by changing a special kind of link in a three-dimensional sphere.

Waldhausen's theorems on topological rigidity

Waldhausen's theorems say that certain three-dimensional spaces are the same if their basic structures are matching.

Smith conjecture

The Smith conjecture says that if a special kind of change is made to a three-dimensional sphere, the places that stay the same cannot form a special kind of knot.

Cyclic surgery theorem

The cyclic surgery theorem gives limits on how changes can be made to certain three-dimensional spaces to keep their basic properties.

Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem

Thurston's theorem says that changing a three-dimensional space in certain ways keeps it with a special kind of shape, as long as some rules are followed.

Thurston's hyperbolization theorem for Haken manifolds

Thurston's theorem says that certain three-dimensional spaces have a special kind of shape inside them.

Tameness conjecture, also called the Marden conjecture or tame ends conjecture

The tameness theorem says that every three-dimensional space with certain properties can be made from a simpler space.

Ending lamination conjecture

The ending lamination theorem says that three-dimensional spaces with certain properties can be described by their shapes and special lines on their edges.

Poincaré conjecture

The Poincaré conjecture is about spaces that look like normal three-dimensional space from close up but might be different overall. It says that if such a space has no edges and every loop can be tightened to a point, it must be a three-dimensional sphere. This was proven by Grigori Perelman.

Thurston's geometrization conjecture

Thurston's geometrization conjecture says that three-dimensional spaces can be broken into pieces, each with a simple geometric shape. This was also proven, building on Perelman's work.

Virtually fibered conjecture and Virtually Haken conjecture

The virtually fibered conjecture and the virtually Haken conjecture are ideas about special properties that three-dimensional spaces might have. Progress has been made on proving these.

Simple loop conjecture

The simple loop conjecture was proven by David Gabai.

Surface subgroup conjecture

The surface subgroup conjecture states that the basic group of certain three-dimensional spaces includes the basic group of a closed surface. Progress has been made on solving this problem.

Important conjectures

Cabling conjecture

The cabling conjecture talks about special ways to change knots in a three-dimensional space. If a certain operation on a knot creates a simpler shape, the conjecture says that the knot must be connected in a special pattern to another knot. This pattern uses two numbers, p and q, to describe how the knots are linked together.