In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ... , a 3-manifold is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ... that locally looks like a
three-dimensional Euclidean space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position of a point. Most commonly, it is the three-dim ... . A 3-
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ... can be thought of as a possible
shape of the universe
In physical cosmology, the shape of the universe refers to both its local and global geometry. Local geometry is defined primarily by its curvature, while the global geometry is characterised by its topology (which itself is constrained by curv ... . Just as a
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ... looks like a
plane
Plane most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
* Plane (mathematics), generalizations of a geometrical plane
Plane or planes may also refer to:
Biology
* Plane ... (a
tangent plane
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ... ) 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
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ... M is a 3-manifold if it is a
second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ... Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ... and if every point in
M has a
neighbourhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ... that is
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ... to
Euclidean 3-space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position of a point. Most commonly, it is the three-dim ... .
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
In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ... ,
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 group ... ,
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ... ,
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ... ,
Teichmüller theory
Teichmüller is a German surname (German for ''pond miller'') and may refer to:
* Anna Teichmüller (1861–1940), German composer
* :de:Frank Teichmüller (19?? – now), former German IG Metall district manager "coast"
* Gustav Teichmüller (1 ... ,
topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants.
While TQFTs were invented by physicists, they are also of mathemati ... ,
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ... ,
Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is an invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer intro ... , and
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ... . 3-manifold theory is considered a part of
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. Representative topics are the theory of 3-manifolds and 4-manifolds, knot theory, ... or
geometric topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topo ... .
A key idea in the theory is to study a 3-manifold by considering special
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ... s 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 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 surface of a suitcase is compressible, because ... and the theory of
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 considers only orientable Haken manifolds, in whi ... s, or one can choose the complementary pieces to be as nice as possible, leading to structures such as
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.
Definitions
Let ''V'' and ''W'' be handlebodies of genus ''g'', and ... s, 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 group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ... s of 3-manifolds strongly reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ... 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
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology (mathematics), homology and cohomology group (mathematics), groups of manifolds. It states that if ''M'' is an ''n''-dim ... and the
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results ... , we have the following
homology group
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ... s:
\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
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ... 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
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type. What this looks like is for a space X there is a list of spaces \_ where\pi_k(X_ ... 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
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ... of
M .
Connected sums
One important topological operation is the
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ... 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
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ... over the real numbers.
3-sphere
A 3-sphere is a higher-dimensional analogue of a sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ... . It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ... . Just as an ordinary sphere (or 2-sphere) is a two-dimensional surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ... that forms the boundary of a ball
A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ... in three dimensions, a 3-sphere is an object with three 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. Thus, a line has a dimension of one (1D) because only one coo ... s 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
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ... of lines passing through the origin 0 in R4 . It is a compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ... , smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ... of dimension ''3'', and is a special case Gr(1, R4 ) of a Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ... space.
RP3 is (diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Defini ... to) 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 \R^3 under the operation of composition.
By definition, a rotation about the origin is a ... , hence admits a group structure; the covering map ''S''3 → RP3 is a map of groups Spin(3) → SO(3), where Spin(3)
In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathbb_2 \to \o ... is a Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ... that is the universal cover
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ... 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
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ... of the integer lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an or ... Z3 (with the action being taken as vector addition). Equivalently, the 3-torus is obtained from the 3-dimensional cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ... by gluing the opposite faces together.
A 3-torus in this sense is an example of a 3-dimensional compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ... manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ... . It is also an example of a compact abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a group ... Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ... . This follows from the fact that the unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ... is a compact abelian Lie group (when identified with the unit complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ... s with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
Hyperbolic 3-space
Hyperbolic space is a homogeneous space
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ... that can be characterized by a constant negative curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ... . It is the model of hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ... . It is distinguished from Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ... s with zero
0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ... curvature that define the Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ... , and models of elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ... (like the 3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ... ) 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
In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ... . Another distinctive property is the amount of space covered by the 3-ball
Three-ball (or "3-ball", colloquially) is a folk game of pool played with any three standard pool s and . The game is frequently gambled upon. The goal is to () the three object balls in as few shots as possible.[exponentially with respect to the radius of the ball, rather than polynomially.
] Poincaré dodecahedral space
The Poincaré
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philos ... homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a spherical 3-manifold
In mathematics, a spherical 3-manifold ''M'' is a 3-manifold of the form
:M=S^3/\Gamma
where \Gamma is a finite subgroup of O(4) acting freely by rotations on the 3-sphere S^3. All such manifolds are prime, orientable, and closed. Spherical 3-ma ... , it is the only homology 3-sphere (besides the 3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ... itself) with a finite fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ... . Its fundamental group is known as the binary icosahedral group
In mathematics, the binary icosahedral group 2''I'' or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 120.
It is an extension of the icosahedral group ''I'' or (2,3,5) o ... and has order 120. This shows the Poincaré conjecture
In the mathematical field of geometric topology, 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.
Originally conjectured b ... cannot be stated in homology terms alone.
In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background
The cosmic microwave background (CMB, CMBR), or relic radiation, is microwave radiation that fills all space in the observable universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dar ... as observed for one year by the WMAP
The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP and Explorer 80), was a NASA spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic mic ... spacecraft led to the suggestion, by Jean-Pierre Luminet
Jean-Pierre Luminet (born 3 June 1951) is a French astrophysicist, specializing in black holes and cosmology. He is an emeritus research director at the CNRS ( Centre national de la recherche scientifique). Luminet is a member of the Laboratoir ... of the Observatoire de Paris
The Paris Observatory (, ), a research institution of the Paris Sciences et Lettres University, is the foremost astronomical observatory of France, and one of the largest astronomical centres in the world. Its historic building is on the Left Ban ... and colleagues, that the shape of the universe
In physical cosmology, the shape of the universe refers to both its local and global geometry. Local geometry is defined primarily by its curvature, while the global geometry is characterised by its topology (which itself is constrained by curv ... is a Poincaré sphere.["Is the universe a dodecahedron?"](_blank) article at PhysicsWorld. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft.
However, there is no strong support for the correctness of the model, as yet.
Seifert–Weber space
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ... , Seifert–Weber space (introduced by Herbert Seifert
Herbert Karl Johannes Seifert (; 27 May 1907, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology.
Biography
Seifert was born in Bernstadt auf dem Eigen, but soon moved to Bautzen, where he atte ... and Constantin Weber) is a closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ... 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. It ... . It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds.
It is constructed by gluing each face of a dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ... to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space. Rotation of 1/10 gives the Poincaré 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\ge 1. That is,
:H_0(X,\Z) = H_n(X,\Z) = \Z
and
:H_i(X,\Z) = \ for all other ''i''.
Therefore ''X'' is a conne ... , and rotation by 5/10 gives 3-dimensional real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properti ... .
With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space
In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1.
It is homogeneous, and satisfies the stronger property of being a symme ... there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold.
It is a quotient space
Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular:
*Quotient space (topology), in case of topological spaces
*Quotient space (linear algebra), in case of vector spaces
*Quotient sp ... of the order-5 dodecahedral honeycomb
In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs) in Hyperbolic space, hyperbolic 3-space. With Schläfli symbol it has f ... , a regular
Regular may refer to:
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings
Other uses
* Regular character, ... tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ... of hyperbolic 3-space
In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1.
It is homogeneous, and satisfies the stronger property of being a symme ... by dodecahedra with this dihedral angle.
Gieseking manifold
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ... , the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is ori ... and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by .
The Gieseking manifold can be constructed by removing the vertices from a tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ... , then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein
David Bernard Alper Epstein (born 1937) is a mathematician known for his work in hyperbolic geometry, 3-manifolds, and group theory, amongst other fields. He co-founded the University of Warwick mathematics department with Christopher Zeeman a ... and Robert C. Penner. Moreover, the angle made by the faces is \pi/3 . The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.
Some important classes of 3-manifolds
* 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 Friedhelm Waldhausen
Friedhelm Waldhausen (born 1938 ...
* 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 considers only orientable Haken manifolds, in whi ...
* 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\ge 1. That is,
:H_0(X,\Z) = H_n(X,\Z) = \Z
and
:H_i(X,\Z) = \ for all other ''i''.
Therefore ''X'' is a conne ... s
* 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. It ...
* I-bundle
In mathematics, an I-bundle is a fiber bundle whose fiber is an interval (mathematics), interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even Line (mathematics)#Ray, rays ... s
* Knot and link complements
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
* 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-manifold case, a lens space can be visualized ...
* Seifert fiber spaces Seifert is a German surname. Notable people with the surname include:
* Alfred Seifert (1850–1901), Czech German painter
* Alfred Seifert (flax miller) (1877–1945), New Zealand flax-miller
* Alwin Seifert (1890–1972), German architect
* Benja ... , Circle bundle
In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S^1.
Oriented circle bundles are also known as principal ''U''(1)-bundles, or equivalently, as principal ''SO''(2)-bundles. In physics, circle bundles are the natural ... s
* Spherical 3-manifold
In mathematics, a spherical 3-manifold ''M'' is a 3-manifold of the form
:M=S^3/\Gamma
where \Gamma is a finite subgroup of O(4) acting freely by rotations on the 3-sphere S^3. All such manifolds are prime, orientable, and closed. Spherical 3-ma ...
* Surface bundles over the circle
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
* 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.
Construction
To obtain a torus bundle: let f be an orientability, orientation-preserv ...
Hyperbolic link complements
A hyperbolic link is a link in the 3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ... with complement
Complement may refer to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class collections into complementary sets
* Complementary color, in the visu ... that has a complete Riemannian metric
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ... of constant negative curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ... , i.e. has a hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ... . A hyperbolic knot is a hyperbolic link with one component
Component may refer to:
In engineering, science, and technology Generic systems
*System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis
* Lumped e ... .
The following examples are particularly well-known and studied.
* Figure eight knot
* Whitehead link
In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop.
Structure
A common way ...
* Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are link (knot theory), topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops wh ...
The classes are not necessarily mutually exclusive.
Some important structures on 3-manifolds
Contact geometry
Contact geometry is the study of a geometric structure on smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ... s given by a hyperplane distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a varia ... in the tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ... and specified by a one-form
In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the to ... , both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'. From the Frobenius theorem , one recognizes the condition as the opposite of the condition that the distribution be determined by a codimension one foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ... on the manifold ('complete integrability').
Contact geometry is in many ways an odd-dimensional counterpart of 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. Symplectic geometry has its origins in the ... , which belongs to the even-dimensional world. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ... , where one can consider either the even-dimensional phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ... of a mechanical system or the odd-dimensional extended phase space that includes the time variable.
Haken manifold
A Haken manifold is a compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ... , P²-irreducible
In mathematics, a P2-irreducible manifold is a 3-manifold that is irreducible and contains no 2-sided \mathbb RP^2 (real projective plane). An orientable manifold is P2-irreducible if and only if it is irreducible.. Every non-orientable P2-irreduc ... 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided 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 surface of a suitcase is compressible, because ... . Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.
A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The 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''. That is, it has a finite cover (a covering s ... asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.
Haken manifolds were introduced by Wolfgang Haken. Haken proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. Jaco and Oertel gave an algorithm to determine if a 3-manifold was Haken.
Essential lamination
An essential lamination is a lamination
Simulated flight (using image stack created by μCT scanning) through the length of a knitting needle that consists of laminated wooden layers: the layers can be differentiated by the change of direction of the wood's vessels
Shattered windshi ... where every leaf is incompressible and end incompressible, if the complementary regions of the lamination are irreducible, and if there are no spherical leaves.
Essential laminations generalize the incompressible surfaces found in Haken manifolds.
Heegaard splitting
A Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies .
Every closed, orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ... three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise . This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory.
Taut foliation
A taut foliation is a 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.
For affine and projective algebraic varieties, the codimension equals ... 1 foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ... of a 3-manifold with the property that there is a single transverse circle intersecting every leaf. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. Equivalently, by a result of Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the Graduate Center of the City University ... , a codimension 1 foliation is taut if there exists a Riemannian metric
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ... that makes each leaf a minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ... .
Taut foliations were brought to prominence by the work of William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurst ... and David Gabai
David Gabai is an American mathematician and the Princeton University Department of Mathematics, Hughes-Rogers Professor of Mathematics at Princeton University. His research focuses on low-dimensional topology and hyperbolic geometry.
Biography ... .
Foundational results
Some results are named as conjectures as a result of historical artifacts.
We begin with the purely topological:
Moise's theorem
In geometric topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topo ... , Moise's theorem, proved by Edwin E. Moise
Edwin Evariste Moise (; December 22, 1918 – December 18, 1998)
was an American mathematician and mathematics education reformer. After his retirement from mathematics he became a literary critic of 19th-century English poetry and had sever ... in, states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows mathematical analysis to be performed on the manifold.
Definition
A smooth structure on a manifold M ... .
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 into two handlebodies.
Definitions
Let ''V'' and ''W'' be handlebodies of genus ''g'', and ... .
Prime decomposition theorem
The prime decomposition theorem for 3-manifolds states that every compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ... , orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ... 3-manifold is the connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ... of a unique (up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ... homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ... ) collection of prime 3-manifold s.
A manifold is ''prime'' if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension.
Kneser–Haken finiteness
Kneser-Haken finiteness says that for each compact 3-manifold, there is a constant C such that any collection of disjoint incompressible embedded surfaces of cardinality greater than C must contain parallel elements.
Loop and Sphere theorems
The loop theorem is a generalization of Dehn's lemma and should more properly be called the "disk theorem". It was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem
In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. I ... .
A simple and useful version of the loop theorem states that if there is a map
:f\colon (D^2,\partial D^2)\to (M,\partial M) \,
with f, \partial D^2 not nullhomotopic in \partial M , then there is an embedding with the same property.
The sphere theorem of gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.
One example is the following:
Let M be an orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ... 3-manifold such that \pi_2(M) is not the trivial group. Then there exists a non-zero element of
\pi_2(M) having a representative that is an embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup.
When some object X is said to be embedded in another object Y ... S^2\to M .
Annulus and Torus theorems
The annulus theorem states that if a pair of disjoint simple closed curves on the boundary of a three manifold are freely homotopic then they cobound a properly embedded annulus. This should not be confused with the high dimensional theorem of the same name.
The torus theorem is as follows: Let M be a compact, irreducible 3-manifold with nonempty boundary. If M admits an essential map of a torus, then M admits an essential embedding of either a torus or an annulus
JSJ decomposition
The JSJ decomposition, also known as the toral decomposition, is a topological
Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ... construct given by the following theorem:
:Irreducible
In philosophy, systems theory, science, and art, emergence occurs when a complex entity has properties or behaviors that its parts do not have on their own, and emerge only when they interact in a wider whole.
Emergence plays a central role ... orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ... closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy ) minimal collection of disjointly embedded incompressible
Incompressible may refer to:
* Incompressible flow, in fluid mechanics
* incompressible vector field, in mathematics
* Incompressible surface, in mathematics
* Incompressible string, in computing
{{Disambig ... tori such that each component of the 3-manifold obtained by cutting along the tori is either 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 embedded, non- boundary parallel, incompressible t ... or Seifert-fibered
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for al ... .
The acronym JSJ is for William Jaco
William Howard Jaco (born July 14, 1940) is an American mathematician who is known for his role in the Jaco–Shalen–Johannson decomposition theorem and efficient triangulations of 3-manifolds. He retired from Oklahoma State University in 2021 ... , Peter Shalen
Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.
Life
He graduated from Stuyvesant High School in 1962, and went on to earn a B.A. from Harvard Coll ... , and Klaus Johannson
Klaus is a German, Dutch and Scandinavian given name and surname. It originated as a short form of Nikolaus, a German form of the Greek given name Nicholas.
Notable persons whose family name is Klaus
* Billy Klaus (1928–2006), American base ... . The first two worked together, and the third worked independently.
Scott core theorem
The Scott core theorem is a theorem about the finite presentability of fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ... s of 3-manifolds due to G. Peter Scott
Godfrey Peter Scott, known as Peter Scott, (1944 – 19 September 2023) was a British-American mathematician, known for the Scott core theorem.
Education and career
He was born in England to Bernard Scott (a mathematician) and Barbara Scott (a ... . The precise statement is as follows:
Given a 3-manifold (not necessarily compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ... ) with finitely generated fundamental group, there is a compact three-dimensional 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 \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ... , called the compact core or Scott core, such that its inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota(x)=x.
An inclusion map may also be referred to as an inclu ... induces an isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ... on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable .
A simplified proof is given in, and a stronger uniqueness statement is proven in.
Lickorish–Wallace theorem
The Lickorish–Wallace theorem states that any closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ... , orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ... , connected 3-manifold may be obtained by performing 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 conceptualized as two steps: ''drilling'' then '' ... on a framed link
In mathematics, a knot is an embedding of the circle () into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation o ... in the 3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ... with \pm 1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.
Waldhausen's theorems on topological rigidity
Friedhelm Waldhausen
Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province, died 2024) was a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory. ... 's theorems on topological rigidity say that certain 3-manifolds (such as those with an incompressible surface) are homeomorphic if there is an isomorphism of fundamental groups which respects the boundary.
Waldhausen conjecture on Heegaard splittings
Waldhausen conjectured that every closed orientable 3-manifold has only finitely many Heegaard splittings (up to homeomorphism) of any given genus.
Smith conjecture
The Smith conjecture (now proven) states that if ''f'' is a diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ... of the 3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ... of finite order , then the fixed point set of ''f'' cannot be a nontrivial knot
A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ... .
Cyclic surgery theorem
The cyclic surgery theorem states that, for a compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ... , connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ... , orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ... , irreducible
In philosophy, systems theory, science, and art, emergence occurs when a complex entity has properties or behaviors that its parts do not have on their own, and emerge only when they interact in a wider whole.
Emergence plays a central role ... three-manifold ''M'' whose boundary is a torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ... ''T'', if ''M'' is not a Seifert-fibered 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 S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for al ... and ''r,s'' are slopes on ''T'' such that their Dehn filling 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 dimension three and is one which distinguishes hyperbol ... s have cyclic fundamental group, then the distance between ''r'' and ''s'' (the minimal number of times that two simple closed curves in ''T'' representing ''r'' and ''s'' must intersect) is at most 1. Consequently, there are at most three Dehn fillings of ''M'' with cyclic fundamental group.
Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem
Thurston's hyperbolic Dehn surgery theorem states: M(u_1, u_2, \dots, u_n) is hyperbolic as long as a finite set of ''exceptional slopes'' E_i is avoided for the ''i''-th cusp for each ''i''. In addition, M(u_1, u_2, \dots, u_n) converges to ''M'' in ''H'' as all p_i^2+q_i^2 \rightarrow \infty for all p_i/q_i corresponding to non-empty Dehn fillings u_i .
This theorem is due to William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurst ... and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in ''H''. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem.
Another important result by Thurston is that volume decreases under hyperbolic Dehn filling. In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic. The proof relies on basic properties of the Gromov norm In the mathematical field of geometric topology, the simplicial volume (also called Gromov norm) is a measure of the topological complexity of a manifold. More generally, the simplicial norm measures the complexity of homology classes.
Given a clo ... .
Jørgensen also showed that the volume function on this space is a continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ... , proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ... function. Thus by the previous results, nontrivial limits in ''H'' are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has ordinal type \omega^\omega . This result is known as the Thurston-Jørgensen theorem. Further work characterizing this set was done by Gromov
Gromov () is a Russian male surname, its feminine counterpart is Gromova (Громова).
Gromov may refer to:
*Alexander Georgiyevich Gromov (born 1947), Russian politician and KGB officer
* Alexander Gromov (born 1959), Russian science fiction ... .
Also, Gabai, Meyerhoff & Milley showed that the 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 surgery, Dehn surgeries on the Whitehead link. It has volume approximately ... has the smallest volume of any closed orientable hyperbolic 3-manifold.
Thurston's hyperbolization theorem for Haken manifolds
One form of Thurston's geometrization theorem states:
If ''M'' is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of ''M'' has a complete hyperbolic structure of finite volume.
The 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 ... implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.
The conditions that the manifold ''M'' should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties. However the condition that the manifold be Haken is unnecessarily strong. Thurston's hyperbolization conjecture states that a closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic, and this follows from Perelman's proof of the Thurston geometrization conjecture.
Tameness conjecture, also called the Marden conjecture or tame ends conjecture
The tameness theorem states that every complete 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. It ... with finitely generated fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ... is topologically tame , in other words homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ... to the interior of a compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ... 3-manifold.
The tameness theorem was conjectured by Marden. It was proved by Agol and, independently, by Danny Calegari
Danny Matthew Cornelius Calegari is a mathematician and, , a professor of mathematics at the University of Chicago. His research interests include geometry, dynamical systems, low-dimensional topology, and geometric group theory.
Education an ... and David Gabai
David Gabai is an American mathematician and the Princeton University Department of Mathematics, Hughes-Rogers Professor of Mathematics at Princeton University. His research focuses on low-dimensional topology and hyperbolic geometry.
Biography ... . It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the 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 independently by and , states that every finitely generated Kleinian group
In mathematics, a Kleinian gr ... and the ending lamination theorem In hyperbolic geometry, the ending lamination theorem, originally conjectured by as the eleventh problem out of his twenty-four questions, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topol ... . It also implies the Ahlfors measure conjecture .
Ending lamination conjecture
The ending lamination theorem, originally conjectured by William Thurston and later proven by Jeffrey Brock
Jeffrey Farlowe Brock (born June 14, 1970, in Bronxville, New York) is an American mathematician, working in low-dimensional geometry and topology. He is known for his contributions to the understanding of hyperbolic 3-manifolds and the geometry ... , Richard Canary
Richard Douglas Canary (born in 1962) is an American mathematician working mainly on low-dimensional topology. He is a professor at the University of Michigan.
Canary obtained his Ph.D. from Princeton University in 1989 under the supervision of ... , and Yair Minsky, states that 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. It ... s with finitely generated fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ... s are determined by their topology together with certain "end invariants", which are geodesic lamination
Simulated flight (using image stack created by μCT scanning) through the length of a knitting needle that consists of laminated wooden layers: the layers can be differentiated by the change of direction of the wood's vessels
Shattered windshi ... s on some surfaces in the boundary of the manifold.
Poincaré conjecture
The 3-sphere is an especially important 3-manifold because of the now-proven Poincaré conjecture
In the mathematical field of geometric topology, 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.
Originally conjectured b ... . Originally conjectured by Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ... , the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ... 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.
After nearly a century of effort by mathematicians, Grigori Perelman
Grigori Yakovlevich Perelman (, ; born 13June 1966) is a Russian mathematician and geometer who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman resigned from his ... presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv
arXiv (pronounced as "archive"—the X represents the Chi (letter), Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not Scholarly pee ... . The proof followed on from the program of Richard S. Hamilton
Richard Streit Hamilton (January 10, 1943 – September 29, 2024) was an American mathematician who served as the Davies Professor of Mathematics at Columbia University.
Hamilton is known for contributions to geometric analysis and partial dif ... to use the Ricci flow
In differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion o ... to attack the problem. Perelman introduced a modification of the standard Ricci flow, called ''Ricci flow with surgery'' to systematically excise singular regions as they develop, in a controlled way. Several teams of mathematicians have verified that Perelman's proof is correct.
Thurston's geometrization conjecture
Thurston's geometrization conjecture states that certain three-dimensional topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ... s each have a unique geometric structure that can be associated with them. It is an analogue of the uniformization theorem
In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generali ... for two-dimensional surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ... s, which states that every simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ... Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ... can be given one of three geometries ( Euclidean , spherical
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ... , or hyperbolic
Hyperbolic may refer to:
* of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics
** Hyperbolic geometry, a non-Euclidean geometry
** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ... ).
In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William , and implies several other conjectures, such as the Poincaré conjecture
In the mathematical field of geometric topology, 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.
Originally conjectured b ... and Thurston's 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.
Relation to other conjectures
A 3-manifold with a Riem ... .
Thurston's 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.
Statement
One form of the hyperbolization theorem state ... implies that 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 considers only orientable Haken manifolds, in whi ... s satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.
Grigori Perelman
Grigori Yakovlevich Perelman (, ; born 13June 1966) is a Russian mathematician and geometer who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman resigned from his ... sketched a proof of the full geometrization conjecture in 2003 using Ricci flow
In differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion o ... with surgery
Surgery is a medical specialty that uses manual and instrumental techniques to diagnose or treat pathological conditions (e.g., trauma, disease, injury, malignancy), to alter bodily functions (e.g., malabsorption created by bariatric surgery s ... .
There are now several different manuscripts (see below) with details of the proof. The Poincaré conjecture and the spherical space form conjecture
In geometric topology, the spherical space form conjecture (now a theorem) states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere.
History
The conjecture was posed by Heinz Hopf in 1926 after dete ... are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.
Virtually fibered conjecture and Virtually Haken conjecture
The virtually fibered conjecture, formulated by American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the "United States" or "America"
** Americans, citizens and nationals of the United States of America
** American ancestry, p ... mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ... William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurst ... , states that every closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ... , irreducible
In philosophy, systems theory, science, and art, emergence occurs when a complex entity has properties or behaviors that its parts do not have on their own, and emerge only when they interact in a wider whole.
Emergence plays a central role ... , 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 embedded, non- boundary parallel, incompressible t ... 3-manifold with infinite fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ... has a finite cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of c ... which is 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 bundles over the circle are a special case of map ... .
The virtually Haken conjecture states that every compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ... , orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ... , irreducible
In philosophy, systems theory, science, and art, emergence occurs when a complex entity has properties or behaviors that its parts do not have on their own, and emerge only when they interact in a wider whole.
Emergence plays a central role ... three-dimensional manifold with infinite fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ... is ''virtually Haken''. That is, it has a finite cover (a covering space
In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In par ... with a finite-to-one covering map) that is a 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 considers only orientable Haken manifolds, in whi ... .
In a posting on the ArXiv on 25 Aug 2009, Daniel Wise implicitly implied (by referring to a then unpublished longer manuscript) that he had proven the Virtually fibered conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken. This was followed by a survey article in Electronic Research Announcements in Mathematical Sciences.
Several more preprints have followed, including the aforementioned longer manuscript by Wise.[Daniel T. Wise, ''The structure of groups with a quasiconvex hierarchy'', https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1] In March 2012, during a conference at 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). It is located in the 5th arrondi ... in Paris, Ian Agol
Ian Agol (; born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds.
Education and career
Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 a ... announced he could prove the 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''. That is, it has a finite cover (a covering s ... for closed hyperbolic 3-manifolds. The proof built on results of Kahn and Markovic in their proof of the Surface subgroup conjecture
In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed manifold, closed, irreducible manifold, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "su ... and results of Wise in proving the Malnormal Special Quotient Theorem and results of Bergeron and Wise for the cubulation of groups. Taken together with Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.
Simple loop conjecture
If f\colon S \rightarrow T is a map of closed connected surfaces such that f_\star \colon \pi_1(S) \rightarrow \pi_1(T) is not injective, then there exists a non-contractible simple closed
curve \alpha \subset S such that f, _a is homotopically trivial. This conjecture was proven by David Gabai
David Gabai is an American mathematician and the Princeton University Department of Mathematics, Hughes-Rogers Professor of Mathematics at Princeton University. His research focuses on low-dimensional topology and hyperbolic geometry.
Biography ... .
Surface subgroup conjecture
The surface subgroup conjecture of Friedhelm Waldhausen
Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province, died 2024) was a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory. ... states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ... has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in 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. Together with Laurent C. Siebenmann he developed the Kirby–Siebenmann invariant ... 's problem list.
Assuming the geometrization conjecture
In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theor ... , the only open case was that of closed 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. It ... s. A proof of this case was announced in the Summer of 2009 by Jeremy Kahn
Jeremy Adam Kahn (born October 26, 1969) is an American mathematician. He works on hyperbolic geometry, Riemann surfaces and complex dynamics.
Education
Kahn grew up in New York City and attended Hunter College High School. He was a child prodi ... and Vladimir Markovic
Vladimir Marković is a Professor of Mathematics at University of Oxford. He was previously the John D. MacArthur Professor at the California Institute of Technology (2013–2020) and Sadleirian Professor of Pure Mathematics at the University ... and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah. A preprint appeared on the arxiv in October 2009. Their paper was published in the Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ... in 2012. In June 2012, Kahn and Markovic were given the Clay Research Award
__NOTOC__
The Clay Research Award is an annual award given by the Oxford-based Clay Mathematics Institute to mathematicians to recognize their achievements in mathematical research. The following mathematicians have received the award:
See als ... s by the Clay Mathematics Institute
The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's sc ... at a ceremony in Oxford
Oxford () is a City status in the United Kingdom, cathedral city and non-metropolitan district in Oxfordshire, England, of which it is the county town.
The city is home to the University of Oxford, the List of oldest universities in continuou ... .
Important conjectures
Cabling conjecture
The cabling conjecture states that if Dehn surgery on a knot in the 3-sphere yields a reducible 3-manifold, then that knot is a (p,q) -cable on some other knot, and the surgery must have been performed using the slope pq .
References
Further reading
*
*
*
*
*
*
*
*
*
External links
*
*Strickland, NeilA Bestiary of Topological Objects
{{authority control
Low-dimensional topology
Geometric topology
HOME
Content is Copyleft Website design, code, and AI is Copyrighted (c) 2014-2017 by Stephen Payne
Consider donating to Wikimedia
As an Amazon Associate I earn from qualifying purchases