In mathematics, a 3-manifold is a 3-dimensional manifold. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Piecewise linear may refer to Piecewise linear function Piecewise linear manifold

Phenomena in three dimensions can be strikingly different from that for other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. Perhaps surprisingly, 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. In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces 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 groups and In Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, given a Riemann surface X, the Teichmüller space of X, notated TX or Teich( X) is a complex A topological quantum field theory (or topological field theory or TQFT) is a Quantum field theory which computes Topological invariants Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Floer homology is a mathematical tool used in the study of Symplectic geometry and low-dimensional Topology. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i 3-manifold theory is considered a part of low-dimensional topology or geometric topology. In Mathematics, low-dimensional topology is the branch of Topology that studies Manifolds of four or fewer dimensions In Mathematics, geometric topology is the study of Manifolds and their Embeddings Low-dimensional topology, concerning questions of dimensions

A key idea in the theory is to study a 3-manifold by considering special surfaces embedded in it. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. 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. In Mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that contains a two-sided Incompressible surface In the mathematical field of Geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it

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). William Paul Thurston (born October 30, 1946) is an American Mathematician. The most prevalent geometry is hyperbolic geometry. In 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. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. Thus, there is an interplay between group theory and topological methods. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups.

## Some famous examples of 3-manifolds

The following examples are particularly well-known and studied. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. This article is about rotations in three-dimensional Euclidean space In Mathematics, real projective space, or RP n is the Projective space of lines in R n +1 In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Mathematics, hyperbolic n -space, denoted H n, is the maximally symmetric Simply connected, n -dimensional In Algebraic topology, a homology sphere is an n -manifold X having the Homology groups of an n - Sphere, for some integer In Mathematics, Seifert -Weber space is a Closed Hyperbolic 3-manifold.

## Some important classes of 3-manifolds

The classes are not necessarily mutually exclusive!

## Foundational results

Some results are named as conjectures as a result of historical artifacts. In Knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four In Knot theory, the Whitehead link, discovered by JHC Whitehead, is one of the most basic links. In Mathematics, the Borromean rings consist of three topological Circles which are linked and form a Brunnian link, i In Topology, a graph manifold (in German Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some Circle bundles They were In Mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that contains a two-sided Incompressible surface In Algebraic topology, a homology sphere is an n -manifold X having the Homology groups of an n - Sphere, for some integer A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant Sectional curvature -1 In Mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a Solid torus A lens space is an example of a Topological space, considered in Mathematics. A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles In Mathematics, a spherical 3-manifold M is a 3-manifold of the form M=S^3/\Gamma where &Gamma is a finite In Mathematics, a surface bundle over the circle is a Fiber bundle with Base space a Circle, and with fiber space a Surface. In Mathematics, in the sub-field of Geometric topology, a torus bundle is a kind of Surface bundle over the circle, which in turn are a class of Three-manifolds In the mathematical field of Geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it In Mathematics, a taut foliation is a Codimension 1 Foliation of a 3-manifold with the property there is a single transverse circle intersecting In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle In Low-dimensional topology, the trigenus is an invariant consisting of a triplet (g_1g_2g_3 assigned to closed 3-manifolds

We begin with the purely topological:

• Moise's theorem - Every 3-manifold has a triangulation, unique up to common subdivision
• As corollary, every compact 3-manifold has a 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
• Prime decomposition theorem
• Kneser-Haken finiteness
• Loop and sphere theorems
• Annulus and Torus theorem
• JSJ decomposition, also known as the toral decomposition
• Scott core theorem
• Lickorish-Wallace theorem
• Waldhausen's theorems on topological rigidity
• Waldhausen conjecture on Heegaard splittings

Theorems where geometry plays an important role in the proof:

Results explicitly linking geometry and topology:

• Thurston's hyperbolic Dehn surgery theorem
• Jorgensen-Thurston theorem that the set of finite volumes of hyperbolic 3-manifolds has order type ωω. In Mathematics, the prime decomposition theorem for 3-manifolds states that every compact orientable 3-manifold is the Connected sum of a unique (up to In the Topology of 3-manifolds the loop theorem is a generalization of Dehn's lemma. In the Topology of 3-manifolds the sphere theorem denotes a family of statements which show us how the image of a 2-sphere under a continuous map into a 3-manifold In Mathematics, the JSJ decomposition, also known as the toral decomposition, is a Topological construct given by the following theorem In Mathematics, the Scott core theorem is a theorem about the finite presentability of Fundamental groups of 3-manifolds due to G In Mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any Closed, Orientable, connected 3-manifold may be In Mathematics, the Smith conjecture was a problem open for many years and proved at the end of the 1970s In Mathematics, hyperbolic Dehn surgery refers to an operation by which one can obtain further Hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold
• Thurston's geometrization theorem for Haken manifolds
• Tameness conjecture, also called the Marden conjecture or tame ends conjecture
• Ending lamination conjecture

## Important conjectures

Some of these are thought to be solved, as of March 2007. Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures In Mathematics, the tameness conjecture states that every complete Hyperbolic 3-manifold with finitely generated Fundamental group is homeomorphic Please see specific articles for more information.

• Thurston's geometrization conjecture
• Virtually fibered conjecture
• Virtually Haken conjecture
• Cabling conjecture
• Surface subgroup conjecture
• Simple loop conjecture
• The smallest hyperbolic 3-manifold is the Weeks manifold. In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among This entry describes the solution of the Poincaré conjecture at a level intended for the general public Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures In the mathematical subfield of 3-manifolds the virtually fibered conjecture of Thurston states every closed irreducible Atoroidal 3-manifold with infinite In Mathematics, the virtually Haken conjecture states that every compact orientable irreducible 3-manifold with infinite fundamental group is virtually Haken In Mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed irreducible 3-manifold with infinite In Mathematics, the Weeks manifold, sometimes called the Fomenko-Matveev-Weeks manifold, is a closed Hyperbolic 3-manifold obtained by (52 and (51
• Lubotzy-Sarnak conjecture on property tau

## References

• Hempel, 3-manifolds, American Mathematical Society, ISBN 0-8218-3695-1
• Jaco, Lectures on three-manifold topology, American Mathematical Society, ISBN 0-8218-1693-4
• Rolfsen, Knots and Links, American Mathematical Society, ISBN 0-914098-16-0
• Thurston, Three-dimensional geometry and topology, Princeton University Press, ISBN 0-691-08304-5
• Adams, The Knot Book, American Mathematical Society, ISBN 0-8050-7380-9
• Hatcher, Notes on basic 3-manifold topology, available online
• R. H. Bing, The Geometric Topology of 3-Manifolds, (1983) American Mathematical Society Colloquium Publications Volume 40, Providence RI, ISBN 0-8218-1040-5.

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