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3-manifold

Index 3-manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. [1]

185 relations: Abelian group, Ahlfors measure conjecture, Annals of Mathematics, ArXiv, Atoroidal, Ball (mathematics), Binary icosahedral group, Borromean rings, Christos Papakyriakopoulos, Circle bundle, Classical mechanics, Clay Mathematics Institute, Clay Research Award, Closed manifold, Codimension, Compact space, Complex number, Conformal map, Connected space, Connected sum, Constant function, Continuous function, Cosmic microwave background, Covering space, Cube, Cubic honeycomb, Curvature, Curvature of Riemannian manifolds, Daniel Wise (mathematician), Danny Calegari, David Gabai, Dehn surgery, Dehn's lemma, Dennis Sullivan, Density theorem for Kleinian groups, Diffeomorphism, Differentiable manifold, Differential form, Dihedral angle, Dimension, Distribution (differential geometry), Dodecahedron, Edwin E. Moise, Elliptic geometry, Embedding, Ending lamination theorem, Euclidean geometry, Euclidean space, Exponential growth, Figure-eight knot (mathematics), ..., Finitely generated group, Fixed point (mathematics), Floer homology, Foliation, Friedhelm Waldhausen, Frobenius theorem (differential topology), Fundamental group, Gauge theory, Generalized Poincaré conjecture, Geometric group theory, Geometric topology, Geometrization conjecture, Graph manifold, Grassmannian, Grigori Perelman, Gromov norm, Group action, Group theory, Haken manifold, Handlebody, Hausdorff space, Heegaard splitting, Henri Poincaré, Herbert Seifert, Homeomorphism, Homogeneous space, Homology sphere, Homotopy, Hyperbolic 3-manifold, Hyperbolic geometry, Hyperbolic space, Hyperbolization theorem, I-bundle, Ian Agol, Inclusion map, Incompressible surface, Institut Henri Poincaré, Irreducibility (mathematics), Isomorphism, Jean-Pierre Luminet, Jeffrey Weeks (mathematician), Knot (mathematics), Knot complement, Knot theory, Lamination (topology), Lattice (group), Lens space, Lie group, Link (knot theory), Low-dimensional topology, Manifold, Mathematician, Mathematics, Meridian (perimetry, visual field), Mikhail Leonidovich Gromov, Minimal surface, Mostow rigidity theorem, Nature (journal), Neighbourhood (mathematics), Number theory, Order (group theory), Order-5 dodecahedral honeycomb, Ordinal number, Orientability, Oxford, P2-irreducible manifold, Pacific Journal of Mathematics, Paris Observatory, Partial differential equation, Path (topology), Peter Shalen, Phase space, Piecewise linear manifold, Plane (geometry), Poincaré conjecture, Presentation of a group, Prime manifold, Proper map, Quotient space (topology), Real projective space, Regular polytope, Ricci flow, Richard S. Hamilton, Riemann surface, Riemannian manifold, Robion Kirby, Rotation group SO(3), Saddle point, Second-countable space, Seifert fiber space, Shape of the universe, Simply connected space, Smale, Smooth structure, Space (mathematics), Sphere, Sphere theorem (3-manifolds), Spherical 3-manifold, Spherical geometry, Spherical space form conjecture, Spin group, Submanifold, Surface (topology), Surface bundle over the circle, Surface subgroup conjecture, Surgery theory, Symplectic geometry, Tame manifold, Tangent bundle, Teichmüller space, Tessellation, Tetrahedron, Three-ball, Three-dimensional space, Thurston elliptization conjecture, Topological quantum field theory, Topological space, Topology, Torus, Torus bundle, Uniformization theorem, Unit circle, United States, Up to, Vector space, Virtually Haken conjecture, Volume form, Weeks manifold, Whitehead link, Wilkinson Microwave Anisotropy Probe, William Jaco, William Thurston, 0, 2-sided, 3-sphere. Expand index (135 more) »

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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Ahlfors measure conjecture

In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0.

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Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

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ArXiv

arXiv (pronounced "archive") is a repository of electronic preprints (known as e-prints) approved for publication after moderation, that consists of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance, which can be accessed online.

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Atoroidal

In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus.

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Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

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Binary icosahedral group

In mathematics, the binary icosahedral group 2I or is a certain nonabelian group of order 120.

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Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link (i.e., removing any ring results in two unlinked rings).

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Christos Papakyriakopoulos

Christos Dimitriou Papakyriakopoulos, commonly known as Papa (Greek: Χρήστος Δημητρίου Παπακυριακόπουλος; June 29, 1914 – June 29, 1976), was a Greek mathematician specializing in geometric topology.

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Circle bundle

In mathematics, a circle bundle is a fiber bundle where the fiber is the circle \scriptstyle \mathbf^1.

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Classical mechanics

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

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Clay Mathematics Institute

The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Peterborough, New Hampshire, United States.

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Clay Research Award

The Clay Research Award is an annual award given by the Oxford-based Clay Mathematics Institute to mathematicians to recognize their achievement in mathematical research.

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Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary.

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Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Conformal map

In mathematics, a conformal map is a function that preserves angles locally.

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Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

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Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds.

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Constant function

In mathematics, a constant function is a function whose (output) value is the same for every input value.

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Cosmic microwave background

The cosmic microwave background (CMB, CMBR) is electromagnetic radiation as a remnant from an early stage of the universe in Big Bang cosmology.

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Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

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Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

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Cubic honeycomb

The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells.

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Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

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Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.

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Daniel Wise (mathematician)

Daniel T. Wise (born January 24, 1971) is an American mathematician who specializes in geometric group theory and 3-manifolds.

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Danny Calegari

Danny M. C. Calegari (born 24 May 1972) is an Australian-American mathematician who is currently a Professor at the University of Chicago.

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David Gabai

David Gabai, a mathematician, is the Hughes-Rogers Professor of Mathematics at Princeton University.

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Dehn surgery

In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds.

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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 map of the disk which is an embedding and is identical to the original on the boundary of the disk.

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Dennis Sullivan

Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician.

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Density theorem for Kleinian groups

In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved by and, states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.

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Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

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Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

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Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.

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Dihedral angle

A dihedral angle is the angle between two intersecting planes.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties.

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Dodecahedron

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.

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Edwin E. Moise

Edwin Evariste Moise (December 22, 1918 – December 18, 1998) was an American mathematician and mathematics education reformer.

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Elliptic geometry

Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.

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Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

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Ending lamination theorem

In hyperbolic geometry, the ending lamination theorem, originally conjectured by, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.

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Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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Exponential growth

Exponential growth is exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function is proportional to the function's current value, resulting in its value at any time being an exponential function of time, i.e., a function in which the time value is the exponent.

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Figure-eight knot (mathematics)

In knot theory, a figure-eight knot (also called Listing's knot or a Cavendish knot) is the unique knot with a crossing number of four.

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Finitely generated group

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.

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Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.

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Floer homology

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology.

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Foliation

In mathematics, a foliation is a geometric tool for understanding manifolds.

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Friedhelm Waldhausen

Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province) is a German mathematician known for his work in algebraic topology.

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Frobenius theorem (differential topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations.

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Fundamental group

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

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Gauge theory

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.

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Generalized Poincaré conjecture

In the mathematical area of topology, the Generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere 'is' a sphere.

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Geometric group theory

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 topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).

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Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

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Geometrization conjecture

In mathematics, Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them.

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Graph manifold

In topology, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles.

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Grassmannian

In mathematics, the Grassmannian is a space which parametrizes all -dimensional linear subspaces of the n-dimensional vector space.

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Grigori Perelman

Grigori Yakovlevich Perelman (a; born 13 June 1966) is a Russian mathematician.

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Gromov norm

In mathematics, the Gromov norm (or simplicial volume) of a compact oriented n-manifold is a norm on the homology (with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle.

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Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

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Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

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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.

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Handlebody

In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces.

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Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

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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 into two handlebodies.

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Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

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Herbert Seifert

Herbert Karl Johannes Seifert (27 May 1907, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology.

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Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

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Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively.

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Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1.

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Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

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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 its sectional curvatures equal to -1.

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Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

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Hyperbolic space

In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

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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.

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I-bundle

In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold.

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Ian Agol

Ian Agol (born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds.

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Inclusion map

In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element, x, of A to x, treated as an element of B: A "hooked arrow" is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (On the other hand, this notation is sometimes reserved for embeddings.) This and other analogous injective functions from substructures are sometimes called natural injections.

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Incompressible surface

In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold.

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Institut Henri Poincaré

The Henri Poincaré Institute (or IHP for Institut Henri Poincaré) is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS).

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Irreducibility (mathematics)

In mathematics, the concept of irreducibility is used in several ways.

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Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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Jean-Pierre Luminet

Jean-Pierre Luminet (born 3 June 1951) is a French astrophysicist, writer and poet, specialized in black holes and cosmology.

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Jeffrey Weeks (mathematician)

Jeffrey Renwick Weeks (born December 10, 1956) is an American mathematician, a geometric topologist and cosmologist.

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Knot (mathematics)

In mathematics, a knot is an embedding of a circle S^1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies).

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Knot complement

In mathematics, the knot complement of a tame knot K is the three-dimensional space surrounding the knot.

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Knot theory

In topology, knot theory is the study of mathematical knots.

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Lamination (topology)

In topology, a branch of mathematics, a lamination is a.

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Lattice (group)

In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.

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Lens space

A lens space is an example of a topological space, considered in mathematics.

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Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

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Link (knot theory)

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together.

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Low-dimensional topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Meridian (perimetry, visual field)

Meridian (plural: "meridians") is used in perimetry and in specifying visual fields.

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Mikhail Leonidovich Gromov

Mikhail Leonidovich Gromov (also Mikhael Gromov, Michael Gromov or Mischa Gromov; Михаи́л Леони́дович Гро́мов; born 23 December 1943), is a French-Russian mathematician known for work in geometry, analysis and group theory.

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Minimal surface

In mathematics, a minimal surface is a surface that locally minimizes its area.

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Mostow rigidity theorem

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique.

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Nature (journal)

Nature is a British multidisciplinary scientific journal, first published on 4 November 1869.

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Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

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Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

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Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

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Order-5 dodecahedral honeycomb

No description.

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Ordinal number

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

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Orientability

In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.

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Oxford

Oxford is a city in the South East region of England and the county town of Oxfordshire.

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P2-irreducible manifold

In mathematics, a P2-irreducible manifold is a 3-manifold that is irreducible and contains no 2-sided \mathbb RP^2 (real projective plane).

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Pacific Journal of Mathematics

The Pacific Journal of Mathematics (ISSN 0030-8730) is a mathematics research journal supported by a number of American, Asian and Australian universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation.

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Paris Observatory

The Paris Observatory (Observatoire de Paris or Observatoire de Paris-Meudon), a research institution of PSL Research University, is the foremost astronomical observatory of France, and one of the largest astronomical centres in the world.

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Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

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Path (topology)

In mathematics, a path in a topological space X is a continuous function f from the unit interval I.

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Peter Shalen

Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology.

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Phase space

In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.

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Piecewise linear manifold

In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it.

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Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

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Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

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Presentation of a group

In mathematics, one method of defining a group is by a presentation.

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Prime manifold

In topology (a mathematical discipline) a prime manifold is an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds.

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Proper map

In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.

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Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

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Real projective space

In mathematics, real projective space, or RPn or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in Rn+1.

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Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.

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Ricci flow

In differential geometry, the Ricci flow (Italian) is an intrinsic geometric flow.

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Richard S. Hamilton

Richard Streit Hamilton (born 1943) is Davies Professor of Mathematics at Columbia University.

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Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.

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Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

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Robion Kirby

Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology.

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Rotation group SO(3)

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.

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Saddle point

In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) of orthogonal function components defining the surface become zero (a stationary point) but are not a local extremum on both axes.

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Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.

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Seifert fiber space

A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles.

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Shape of the universe

The shape of the universe is the local and global geometry of the universe.

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Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

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Smale

Smale is a surname.

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Smooth structure

In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function.

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Space (mathematics)

In mathematics, a space is a set (sometimes called a universe) with some added structure.

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Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

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Sphere theorem (3-manifolds)

In mathematics, in the topology of 3-manifolds, the sphere theorem of gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.

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Spherical 3-manifold

In mathematics, a spherical 3-manifold M is a 3-manifold of the form where \Gamma is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere S^3.

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Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere.

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Spherical space form conjecture

In geometric topology, the spherical space form conjecture states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere.

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Spin group

In mathematics the spin group Spin(n) is the double cover of the special orthogonal group, such that there exists a short exact sequence of Lie groups (with) As a Lie group, Spin(n) therefore shares its dimension,, and its Lie algebra with the special orthogonal group.

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Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.

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Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.

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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.

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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.

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Surgery theory

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by.

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Symplectic geometry

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

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Tame manifold

In geometry, a tame manifold is a manifold with a well-behaved compactification.

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Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.

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Teichmüller space

In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.

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Tessellation

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.

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Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

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Three-ball

Three-ball (or "3-ball", colloquially) is a folk game of pool played with any three standard pool and.

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Three-dimensional space

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).

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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.

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Topological quantum field theory

A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

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Torus bundle

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.

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Uniformization theorem

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.

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Unit circle

In mathematics, a unit circle is a circle with a radius of one.

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United States

The United States of America (USA), commonly known as the United States (U.S.) or America, is a federal republic composed of 50 states, a federal district, five major self-governing territories, and various possessions.

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Up to

In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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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.

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Volume form

In mathematics, a volume form on a differentiable manifold is a top-dimensional form (i.e., a differential form of top degree).

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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.

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Whitehead link

In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links.

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Wilkinson Microwave Anisotropy Probe

The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP), was a spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic microwave background (CMB) – the radiant heat remaining from the Big Bang.

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William Jaco

Dr.

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William Thurston

William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.

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0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

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2-sided

In topology, a compact codimension one submanifold F of a manifold M is said to be 2-sided in M when there is an embedding with h(x,0).

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3-sphere

In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.

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3-dimensional topology, 3-manifolds, Three-dimensional manifold, Three-manifold, Three-manifolds.

References

[1] https://en.wikipedia.org/wiki/3-manifold

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