# Category: 3-manifolds

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 approx
Eventually (mathematics)
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered i
Thurston norm
In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of hom
Heegaard splitting
In the mathematical field of geometric topology, a Heegaard splitting (Danish: [ˈhe̝ˀˌkɒˀ]) is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Loop theorem
In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma a
Surface bundle over the circle
In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundle
Torus bundle
A torus bundle, in the sub-field of geometric topology in mathematics, is a kind of surface bundle over the circle, which in turn is a class of three-manifolds.
Lickorish–Wallace theorem
In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3
Incompressible surface
In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a "nontrivial" surface that cannot be simplified. In non-mathematical terms, the
SnapPea
SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first version
(−2,3,7) pretzel knot
In geometric topology, a branch of mathematics, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after Ron Fintushel and Ronald J. Stern), is an important example of a pretzel k
2π theorem
In mathematics, the 2π theorem of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold. Let M be a cusped hy
Scott core theorem
In mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott,. The precise statement is as follows: Given a 3-manifold
Figure-eight knot (mathematics)
In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the u
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-di
Ending lamination theorem
In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their
Unknotting problem
In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algori
Lens space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-ma
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 consid
Property P conjecture
In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obt
Virtually fibered conjecture
In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with i
Hyperbolization theorem
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.
In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a
Geometric topology (object)
In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume.
Analytic torsion
In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dim
Normal surface
In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad (see figure). A triangle
Surface subgroup conjecture
In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup
Horosphere
In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one sid
Open book decomposition
In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold M into a union of surfaces (necessarily with boundary) and solid tori. Open books
JSJ decomposition
In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: Irreducible orientable closed (i.e., compact and without bounda
Smith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot. Paul A. Smith showed that a non-tr
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group of th
Atoroidal
In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus.There are two major variations in this terminology: an essential torus may be defined geometrically, as an embed
Clasper (mathematics)
In the mathematical field of low-dimensional topology, a clasper is a surface (with extra structure) in a 3-manifold on which surgery can be performed.
Solid torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product of the disk and the circle, endowed with the product topol
3-torus
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, In contrast, the usual torus is the Cartesian product of on
P2-irreducible manifold
In mathematics, a P2-irreducible manifold is a 3-manifold that is irreducible and contains no 2-sided (real projective plane). An orientable manifold is P2-irreducible if and only if it is irreducible
Hantzsche–Wendt manifold
The Hantzsche–Wendt manifold, also known as the HW manifold or didicosm, is a compact, orientable, flat 3-manifold, first studied by and Hilmar Wendt in 1934. It is the only closed flat 3-manifold wit
Prime decomposition of 3-manifolds
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 homeomorphism) finite collection of prime 3-man
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a -bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are
Dehn's lemma
In mathematics, Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disk's interior, implies the existence of another piecewise-linear m
Tameness theorem
In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a
Spherical 3-manifold
In mathematics, a spherical 3-manifold M is a 3-manifold of the form where is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere . All such manifolds are prime, orientable, and clos
Graph manifold
In topology, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles. They were discovered and classified by the German topologist Friedhe
Trigenus
In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple . It is obtained by minimizing the genera of three orientable handle bodies — with no i
I-bundle
In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can
3-manifold
In mathematics, a 3-manifold is a space 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 looks like a plane t
Poincaré conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (UK: /ˈpwæ̃kæreɪ/, US: /ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is th
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to J. H. C. Whitehead discovered this puzzling object while he was trying to prove the Poincaré
Thurston elliptization conjecture
William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature.
Hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all
Berge knot
In the mathematical theory of knots, a Berge knot (named after mathematician John Berge) or doubly primitive knot is any member of a particular family of knots in the 3-sphere. A Berge knot K is defin
Finite type invariant
In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant (so named after Victor Anatolyevich Vassiliev), is a knot invariant that can be extended (in a precise manner to be
Virtually Haken conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. Th
Compression body
In the theory of 3-manifolds, a compression body is a kind of generalized handlebody. A compression body is either a handlebody or the result of the following construction: Let be a compact, closed su
Branched surface
In mathematics, a branched surface is a generalization of both surfaces and train tracks.
Homology sphere
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer . That is, and for all other i. Therefore X is a connected space, with one non-z
Picard horn
A Picard horn, also called the Picard topology or Picard model, is one of the oldest known hyperbolic 3-manifolds, first described by Émile Picard in 1884. The manifold is the quotient of the upper ha
Weeks manifold
In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has vo
Handle decompositions of 3-manifolds
In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study.
In the mathematical theory of knots, a pretzel link is a special kind of link. It consists of a finite number tangles made of two intertwined circular helices, The tangles are connected cyclicly, the
Algebraic topology (object)
In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g)
The geometry and topology of three-manifolds
The geometry and topology of three-manifolds is a set of widely circulated but unpublished notes by William Thurston from 1978 to 1980 describing his work on 3-manifolds. The notes introduced several
Cyclic surgery theorem
In three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is
Geometrization conjecture
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue
Gordon–Luecke theorem
In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between
Hyperbolic Dehn surgery
In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimensi
Ehrenpreis conjecture
In mathematics, the Ehrenpreis conjecture of Leon Ehrenpreis states that for any K greater than 1, any two closed Riemann surfaces of genus at least 2 have finite-degree covers which are K-quasiconfor
Meyerhoff manifold
In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by surgery on the figure-8 knot complement. It was introduced by Robert Meyerhoff as a possible candidat
Solid Klein bottle
In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. It is homeomorphic to the quotient space obtained by gluing the top dis
Arithmetic topology
Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.
Ricci flow
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow (/ˈriːtʃi/ REE-chee, Italian: [ˈrittʃi]), sometimes also referred to as Hamilton's Ricci flow, is a certain p
Arithmetic hyperbolic 3-manifold
In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are part
Pleated surface
In geometry, a pleated surface is roughly a surface that may have simple folds but is not crumpled in more complicated ways. More precisely, a pleated surface is an isometry from a complete hyperbolic
Seifert–Weber space
In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahe
Bing's recognition theorem
In topology, a branch of mathematics, Bing's recognition theorem, named for R. H. Bing, asserts that a necessary and sufficient condition for a 3-manifold M to be homeomorphic to the 3-sphere is that
Sphere theorem (3-manifolds)
In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedde
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conc