In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s has a unique
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
structure that can be associated with it. It is an analogue of the
uniformization theorem
In mathematics, the uniformization theorem says 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 generalization o ...
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 between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
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 () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
, or
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
).
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
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by , and implies several other conjectures, such as the
Poincaré conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
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 Thurston's geometrization theor ...
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 w ...
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 ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
announced a proof of the full geometrization conjecture in 2003 using
Ricci flow
In the mathematical fields of 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 analo ...
with
surgery
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pat ...
in two papers posted at the arxiv.org preprint server. Perelman's papers were studied by several independent groups that produced books and online manuscripts filling in the complete details of his arguments. Verification was essentially complete in time for Perelman to be awarded the 2006
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
for his work, and in 2010 the
Clay Mathematics Institute
The Clay Mathematics Institute (CMI) is a private, non-profit foundation (nonprofit), foundation dedicated to increasing and disseminating mathematics, mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address i ...
awarded him its 1 million USD prize for solving the Poincare conjecture, though Perelman declined to accept either award.
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 de ...
are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.
The conjecture
A 3-manifold is called closed if it is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
and has no
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
.
Every closed 3-manifold has a
prime decomposition
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are suf ...
: this means it 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
prime 3-manifolds (this decomposition is essentially unique except for a small problem in the case of
non-orientable manifolds). This reduces much of the study of 3-manifolds to the case of prime 3-manifolds: those that cannot be written as a non-trivial connected sum.
Here is a statement of Thurston's conjecture:
:Every oriented prime closed
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
can be cut along
tori, so that the
interior of each of the resulting manifolds has a geometric structure with finite volume.
There are 8 possible geometric structures in 3 dimensions, described in the next section. There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are
Seifert manifold
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 ...
s or
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 ...
called the
JSJ decomposition In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
: Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isot ...
, which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. (For example, the mapping torus of an
Anosov map
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...
of a torus has a finite volume solv structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.)
For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the
oriented double cover. It is also possible to work directly with non-orientable manifolds, but this gives some extra complications: it may be necessary to cut along
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
s and
Klein bottle
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
s as well as spheres and tori, and manifolds with a projective plane boundary component usually have no geometric structure.
In 2 dimensions the analogous statement says that every surface (without boundary) has a geometric structure consisting of a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
with constant curvature; it is not necessary to cut the manifold up first.
The eight Thurston geometries
A model geometry is a simply connected smooth manifold ''X'' together with a
transitive action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
''G'' on ''X'' with compact stabilizers.
A model geometry is called maximal if ''G'' is maximal among groups acting smoothly and transitively on ''X'' with compact stabilizers. Sometimes this condition is included in the definition of a model geometry.
A geometric structure on a manifold ''M'' is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
from ''M'' to ''X''/Γ for some model geometry ''X'', where Γ is a
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
subgroup of ''G'' acting freely on ''X'' ; this is a special case of a complete
(''G'',''X'')-structure. If a given manifold admits a geometric structure, then it admits one whose model is maximal.
A 3-dimensional model geometry ''X'' is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on ''X''. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries. (There are also
uncountably
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many Element (mathematics), elements to be countable set, countable. The uncountability of a set is closely related to its cardinal number: a se ...
many model geometries without compact quotients.)
There is some connection with the
Bianchi groups: the 3-dimensional Lie groups. Most Thurston geometries can be realized as a left invariant metric on a Bianchi group. However
S2 × R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of
solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.
Spherical geometry S3
The point stabilizer is O(3, R), and the group ''G'' is the 6-dimensional Lie group O(4, R), with 2 components. The corresponding manifolds are exactly the closed 3-manifolds with finite
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
. Examples include the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
, the
Poincaré homology sphere,
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 visualize ...
s. This geometry can be modeled as a left invariant metric on the
Bianchi group of type IX. Manifolds with this geometry are all compact, orientable, and have the structure of a
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
(often in several ways). The complete list of such manifolds is given in the article on
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 SO(4) acting freely by rotations on the 3-sphere S^3. All such manifolds are prime, orientable, and closed. Spherical 3-ma ...
s. Under Ricci flow, manifolds with this geometry collapse to a point in finite time.
Euclidean geometry ''E''3
The point stabilizer is O(3, R), and the group ''G'' is the 6-dimensional Lie group R
3 × O(3, R), with 2 components. Examples are the
3-torus
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, \mathbb^3 = S^1 \times S^1 \times S^1. In contrast, the usual torus is the Cartesian product of only two ...
, and more generally the
mapping torus In mathematics, the mapping torus in topology of a homeomorphism ''f'' of some topological space ''X'' to itself is a particular geometric construction with ''f''. Take the cartesian product of ''X'' with a closed interval ''I'', and glue the boun ...
of a finite-order
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
of the 2-torus; see
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 orientation-preserving homeomor ...
. There are exactly 10 finite closed 3-manifolds with this geometry, 6 orientable and 4 non-orientable. This geometry can be modeled as a left invariant metric on the
Bianchi groups of type I or VII0. Finite volume manifolds with this geometry are all compact, and have the structure of a
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
(sometimes in two ways). The complete list of such manifolds is given in the article on
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s. Under Ricci flow, manifolds with Euclidean geometry remain invariant.
Hyperbolic geometry H3
The point stabilizer is O(3, R), and the group ''G'' is the 6-dimensional Lie group O
+(1, 3, R), with 2 components. There are enormous numbers of examples of these, and their classification is not completely understood. The example with smallest volume is 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 surgeries on the Whitehead link. It has volume approximately equal to 0.942 ...
. Other examples are given by the
Seifert–Weber space In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discover ...
, or "sufficiently complicated"
Dehn surgeries on
links, or most
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 w ...
s. The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible,
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 ...
, and has infinite fundamental group. This geometry can be modeled as a left invariant metric on the
Bianchi group of type V or VIIh≠0. Under Ricci flow, manifolds with hyperbolic geometry expand.
The geometry of S2 × R
The point stabilizer is O(2, R) × Z/2Z, and the group ''G'' is O(3, R) × R × Z/2Z, with 4 components. The four finite volume manifolds with this geometry are: S
2 × S
1, the mapping torus of the antipode map of S
2, the connected sum of two copies of 3-dimensional projective space, and the product of S
1 with two-dimensional projective space. The first two are mapping tori of the identity map and antipode map of the 2-sphere, and are the only examples of 3-manifolds that are prime but not irreducible. The third is the only example of a non-trivial connected sum with a geometric structure. This is the only model geometry that cannot be realized as a left invariant metric on a 3-dimensional Lie group. Finite volume manifolds with this geometry are all compact and have the structure of a
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
(often in several ways). Under normalized Ricci flow manifolds with this geometry converge to a 1-dimensional manifold.
The geometry of H2 × R
The point stabilizer is O(2, R) × Z/2Z, and the group ''G'' is O
+(1, 2, R) × R × Z/2Z, with 4 components. Examples include the product of a
hyperbolic surface
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
with a circle, or more generally the mapping torus of an isometry of a hyperbolic surface. Finite volume manifolds with this geometry have the structure of a
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
if they are orientable. (If they are not orientable the natural fibration by circles is not necessarily a Seifert fibration: the problem is that some fibers may "reverse orientation"; in other words their neighborhoods look like fibered solid Klein bottles rather than solid tori.) The classification of such (oriented) manifolds is given in the article on
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s. This geometry can be modeled as a left invariant metric on the
Bianchi group of type III. Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.
The geometry of the universal cover of SL(2, "R")
The
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
of
SL(2, R) is denoted
. It fibers over H
2, and the space is sometimes called "Twisted H
2 × R". The group ''G'' has 2 components. Its identity component has the structure
. The point stabilizer is O(2,R).
Examples of these manifolds include: the manifold of unit vectors of the tangent bundle of a hyperbolic surface, and more generally the
Brieskorn homology spheres (excepting the 3-sphere and the
Poincare dodecahedral space). This geometry can be modeled as a left invariant metric on the
Bianchi group of type VIII or III. Finite volume manifolds with this geometry are orientable and have the structure of a
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
. The classification of such manifolds is given in the article on
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s. Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.
Nil geometry
This fibers over ''E''
2, and so is sometimes known as "Twisted ''E''
2 × R". It is the geometry of the
Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ' ...
. The point stabilizer is O(2, R). The group ''G'' has 2 components, and is a semidirect product of the 3-dimensional Heisenberg group by the group O(2, R) of isometries of a circle. Compact manifolds with this geometry include the mapping torus of a
Dehn twist
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).
Definition
Suppose that ''c'' is a simple closed curve in a closed, orientable surface ''S''. Let ' ...
of a 2-torus, or the quotient of the Heisenberg group by the "integral Heisenberg group". This geometry can be modeled as a left invariant metric on the
Bianchi group of type II. Finite volume manifolds with this geometry are compact and orientable and have the structure of a
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
. The classification of such manifolds is given in the article on
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s. Under normalized Ricci flow, compact manifolds with this geometry converge to R
2 with the flat metric.
Sol geometry
This geometry (also called Solv geometry) fibers over the line with fiber the plane, and is the geometry of the identity component of the group ''G''. The point stabilizer is the dihedral group of order 8. The group ''G'' has 8 components, and is the group of maps from 2-dimensional Minkowski space to itself that are either isometries or multiply the metric by −1. The identity component has a normal subgroup R
2 with quotient R, where R acts on R
2 with 2 (real) eigenspaces, with distinct real eigenvalues of product 1. This is the
Bianchi group of type VI0 and the geometry can be modeled as a left invariant metric on this group. All finite volume manifolds with solv geometry are compact. The compact manifolds with solv geometry are either the
mapping torus In mathematics, the mapping torus in topology of a homeomorphism ''f'' of some topological space ''X'' to itself is a particular geometric construction with ''f''. Take the cartesian product of ''X'' with a closed interval ''I'', and glue the boun ...
of an
Anosov map
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...
of the 2-torus (such a map is an automorphism of the 2-torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, such as
), or quotients of these by groups of order at most 8. The eigenvalues of the automorphism of the torus generate an order of a real quadratic field, and the solv manifolds can be classified in terms of the units and ideal classes of this order.
Under normalized Ricci flow compact manifolds with this geometry converge (rather slowly) to R
1.
Uniqueness
A closed 3-manifold has a geometric structure of at most one of the 8 types above, but finite volume non-compact 3-manifolds can occasionally have more than one type of geometric structure. (Nevertheless, a manifold can have many different geometric structures of the same type; for example, a surface of genus at least 2 has a continuum of different hyperbolic metrics.) More precisely, if ''M'' is a manifold with a finite volume geometric structure, then the type of geometric structure is almost determined as follows, in terms of the fundamental group π
1(''M''):
*If π
1(''M'') is finite then the geometric structure on ''M'' is spherical, and ''M'' is compact.
*If π
1(''M'') is virtually cyclic but not finite then the geometric structure on ''M'' is S
2×R, and ''M'' is compact.
*If π
1(''M'') is virtually abelian but not virtually cyclic then the geometric structure on ''M'' is Euclidean, and ''M'' is compact.
*If π
1(''M'') is virtually nilpotent but not virtually abelian then the geometric structure on ''M'' is nil geometry, and ''M'' is compact.
*If π
1(''M'') is virtually solvable but not virtually nilpotent then the geometric structure on ''M'' is solv geometry, and ''M'' is compact.
*If π
1(''M'') has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on ''M'' is either H
2×R or the universal cover of SL(2, R). The manifold ''M'' may be either compact or non-compact. If it is compact, then the 2 geometries can be distinguished by whether or not π
1(''M'') has a finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
subgroup that splits as a semidirect product of the normal cyclic subgroup and something else. If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples (such as the complement of a trefoil knot) where a manifold may have a finite volume geometric structure of either type.
*If π
1(''M'') has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on ''M'' is hyperbolic, and ''M'' may be either compact or non-compact.
Infinite volume manifolds can have many different types of geometric structure: for example, R
3 can have 6 of the different geometric structures listed above, as 6 of the 8 model geometries are homeomorphic to it. Moreover if the volume does not have to be finite there are an infinite number of new geometric structures with no compact models; for example, the geometry of almost any non-unimodular 3-dimensional Lie group.
There can be more than one way to decompose a closed 3-manifold into pieces with geometric structures. For example:
*Taking connected sums with several copies of S
3 does not change a manifold.
*The connected sum of two projective 3-spaces has a S
2×R geometry, and is also the connected sum of two pieces with S
3 geometry.
*The product of a surface of negative curvature and a circle has a geometric structure, but can also be cut along tori to produce smaller pieces that also have geometric structures. There are many similar examples for Seifert fiber spaces.
It is possible to choose a "canonical" decomposition into pieces with geometric structure, for example by first cutting the manifold into prime pieces in a minimal way, then cutting these up using the smallest possible number of tori. However this minimal decomposition is not necessarily the one produced by Ricci flow; in fact, the Ricci flow can cut up a manifold into geometric pieces in many inequivalent ways, depending on the choice of initial metric.
History
The
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for
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 w ...
s.
In 1982,
Richard S. Hamilton showed that given a closed 3-manifold with a metric of positive
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
, the
Ricci flow
In the mathematical fields of 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 analo ...
would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes "almost round" just before the collapse. He later developed a program to prove the geometrization conjecture by
Ricci flow with surgery Ricci () is an Italian surname, derived from the adjective "riccio", meaning curly. Notable Riccis Arts and entertainment
* Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin
* Christina Ricci (born 1980), Ameri ...
. The idea is that the Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology of the manifold. Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the "positive curvature" geometries S
3 and S
2 × R, while what is left at large times should have a
thick–thin decomposition into a "thick" piece with hyperbolic geometry and a "thin"
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 in 1967. This definition allows a ...
.
In 2003,
Grigori Perelman
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
announced a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above.
One component of Perelman's proof was a novel collapsing theorem in Riemannian geometry. Perelman did not release any details on the proof of this result (Theorem 7.4 in the preprint 'Ricci flow with surgery on three-manifolds'). Beginning with Shioya and Yamaguchi, there are now several different proofs of Perelman's collapsing theorem, or variants thereof. Shioya and Yamaguchi's formulation was used in the first fully detailed formulations of Perelman's work.
A second route to the last part of Perelman's proof of geometrization is the method of
Bessières ''et al.'', which uses Thurston's hyperbolization theorem for Haken manifolds and
Gromov
Gromov (russian: Громов) 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), R ...
's
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
for 3-manifolds. A book by the same authors with complete details of their version of the proof has been published by the
European Mathematical Society
The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ...
.
[L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010. Available at https://www-fourier.ujf-grenoble.fr/~besson/book.pdf]
Notes
References
*L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010.
*M. Boilea
Geometrization of 3-manifolds with symmetries*F. Bonahon ''Geometric structures on 3-manifolds'' Handbook of Geometric Topology (2002) Elsevier.
*
* Allen Hatcher
''Notes on Basic 3-Manifold Topology''2000
*J. Isenberg, M. Jackson, ''Ricci flow of locally homogeneous geometries on a Riemannian manifold'', J. Diff. Geom. 35 (1992) no. 3 723–741.
*
*
John Morgan (mathematician), John W. Morgan''Recent progress on the Poincaré conjecture and the classification of 3-manifolds.''Bulletin Amer. Math. Soc. 42 (2005) no. 1, 57–78 (expository article explains the eight geometries and geometrization conjecture briefly, and gives an outline of Perelman's proof of the Poincaré conjecture)
*
*
*
*
*
*
Scott, Peterbr>
''The geometries of 3-manifolds.''errata Bull. London Math. Soc. 15 (1983), no. 5, 401–487.
* This gives the original statement of the conjecture.
* William Thurston. ''Three-dimensional geometry and topology. Vol. 1''. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. (in depth explanation of the eight geometries and the proof that there are only eight)
* William Thurston
The Geometry and Topology of Three-Manifolds 1980 Princeton lecture notes on geometric structures on 3-manifolds.
External links
* A public lecture on the Poincaré and geometrization conjectures, given by
C. McMullen at Harvard in 2006.
{{DEFAULTSORT:Geometrization Conjecture
Geometric topology
Riemannian geometry
3-manifolds
Conjectures