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97-405: 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 Thurston  ( 1982 ), and implies several other conjectures, such as

194-410: A transitive action of a Lie group 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 from M to X /Γ for some model geometry X , where Γ

291-432: A 1-dimensional manifold. The point stabilizer is O(2, R ) × Z /2 Z , and the group G is O(1, 2, R ) × R × Z /2 Z , with 4 components. Examples include the product of a hyperbolic surface 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 if they are orientable. (If they are not orientable

388-628: A 2-dimensional manifold. The universal cover of SL(2, R ) is denoted S L ~ ( 2 , R ) {\displaystyle {\widetilde {\rm {SL}}}(2,\mathbf {R} )} . It fibers over H , and the space is sometimes called "Twisted H × R". The group G has 2 components. Its identity component has the structure ( R × S L ~ 2 ( R ) ) / Z {\displaystyle (\mathbf {R} \times {\widetilde {\rm {SL}}}_{2}(\mathbf {R} ))/\mathbf {Z} } . The point stabilizer

485-487: A bijection. If D f x {\displaystyle Df_{x}} is a bijection at x {\displaystyle x} then f {\displaystyle f} is said to be a local diffeomorphism (since, by continuity, D f y {\displaystyle Df_{y}} will also be bijective for all y {\displaystyle y} sufficiently close to x {\displaystyle x} ). Given

582-432: A compact subset of M {\displaystyle M} , this follows by fixing a Riemannian metric on M {\displaystyle M} and using the exponential map for that metric. If r {\displaystyle r} is finite and the manifold is compact, the space of vector fields is a Banach space . Moreover, the transition maps from one chart of this atlas to another are smooth, making

679-464: A differentiable manifold that is second-countable and Hausdorff . The diffeomorphism group of M {\displaystyle M} is the group of all C r {\displaystyle C^{r}} diffeomorphisms of M {\displaystyle M} to itself, denoted by Diff r ( M ) {\displaystyle {\text{Diff}}^{r}(M)} or, when r {\displaystyle r}

776-567: A doctorate in mathematics from the University of California, Berkeley under Morris Hirsch , with his thesis Foliations of Three-Manifolds which are Circle Bundles in 1972. After completing his Ph.D., Thurston spent a year at the Institute for Advanced Study , then another year at the Massachusetts Institute of Technology as an assistant professor. In 1974, Thurston was appointed

873-505: A full professor at Princeton University . He returned to Berkeley in 1991 to be a professor (1991-1996) and was also director of the Mathematical Sciences Research Institute (MSRI) from 1992 to 1997. He was on the faculty at UC Davis from 1996 until 2003, when he moved to Cornell University . Thurston was an early adopter of computing in pure mathematics research. He inspired Jeffrey Weeks to develop

970-482: A homeomorphism, f {\displaystyle f} and its inverse need only be continuous . Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism. f : M → N {\displaystyle f:M\to N} is a diffeomorphism if, in coordinate charts , it satisfies the definition above. More precisely: Pick any cover of M {\displaystyle M} by compatible coordinate charts and do

1067-475: 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 ): Infinite volume manifolds can have many different types of geometric structure: for example, R can have 6 of

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1164-531: A mathematician at Indiana University Bloomington . Thurston had two children with his second wife, Julian Muriel Thurston: Hannah Jade and Liam. Thurston died on August 21, 2012, in Rochester, New York , of a sinus mucosal melanoma that was diagnosed in 2011. 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

1261-404: A projective plane boundary component usually have no geometric structure. In 2 dimensions, every closed surface has a geometric structure consisting of a metric with constant curvature; it is not necessary to cut the manifold up first. Specifically, every closed surface is diffeomorphic to a quotient of S , E , or H . A model geometry is a simply connected smooth manifold X together with

1358-472: A seminal contribution to the research literature". He was awarded the 2012 Leroy P. Steele Prize by the American Mathematical Society for seminal contribution to research. The citation described his work as having "revolutionized 3-manifold theory". Thurston and his first wife, Rachel Findley, had three children: Dylan, Nathaniel, and Emily. Dylan was a MOSP participant (1988–90) and is

1455-492: 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: 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 manifolds or atoroidal called

1552-542: A smooth map from dimension n {\displaystyle n} to dimension k {\displaystyle k} , if D f {\displaystyle Df} (or, locally, D f x {\displaystyle Df_{x}} ) is surjective, f {\displaystyle f} is said to be a submersion (or, locally, a "local submersion"); and if D f {\displaystyle Df} (or, locally, D f x {\displaystyle Df_{x}} )

1649-434: A stress-induced transformation is called a deformation and may be described by a diffeomorphism. A diffeomorphism f : U → V {\displaystyle f:U\to V} between two surfaces U {\displaystyle U} and V {\displaystyle V} has a Jacobian matrix D f {\displaystyle Df} that is an invertible matrix . In fact, it

1746-442: 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 planes and Klein bottles as well as spheres and tori, and manifolds with

1843-477: A well-defined inverse if and only if D f x {\displaystyle Df_{x}} is a bijection. The matrix representation of D f x {\displaystyle Df_{x}} is the n × n {\displaystyle n\times n} matrix of first-order partial derivatives whose entry in the i {\displaystyle i} -th row and j {\displaystyle j} -th column

1940-442: Is σ {\displaystyle \sigma } -compact, there is a sequence of compact subsets K n {\displaystyle K_{n}} whose union is M {\displaystyle M} . Then: The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of C r {\displaystyle C^{r}} vector fields ( Leslie 1967 ). Over

2037-921: Is ∂ f i / ∂ x j {\displaystyle \partial f_{i}/\partial x_{j}} . This so-called Jacobian matrix is often used for explicit computations. Diffeomorphisms are necessarily between manifolds of the same dimension . Imagine f {\displaystyle f} going from dimension n {\displaystyle n} to dimension k {\displaystyle k} . If n < k {\displaystyle n<k} then D f x {\displaystyle Df_{x}} could never be surjective, and if n > k {\displaystyle n>k} then D f x {\displaystyle Df_{x}} could never be injective. In both cases, therefore, D f x {\displaystyle Df_{x}} fails to be

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2134-401: Is Then the image ( d u , d v ) = ( d x , d y ) D f {\displaystyle (du,dv)=(dx,dy)Df} is a linear transformation , fixing the origin, and expressible as the action of a complex number of a particular type. When ( dx ,  dy ) is also interpreted as that type of complex number, the action is of complex multiplication in

2231-464: Is multiply transitive ( Banyaga 1997 , p. 29). In 1926, Tibor Radó asked whether the harmonic extension of any homeomorphism or diffeomorphism of the unit circle to the unit disc yields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards by Hellmuth Kneser . In 1945, Gustave Choquet , apparently unaware of this result, produced a completely different proof. The (orientation-preserving) diffeomorphism group of

2328-535: Is simply connected , a differentiable map f : U → V {\displaystyle f:U\to V} is a diffeomorphism if it is proper and if the differential D f x : R n → R n {\displaystyle Df_{x}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} is bijective (and hence a linear isomorphism ) at each point x {\displaystyle x} in U {\displaystyle U} . Some remarks: It

2425-409: Is O(2, R ) × Z /2 Z , and the group G is O(3, R ) × R × Z /2 Z , with 4 components. The four finite volume manifolds with this geometry are: S × S , the mapping torus of the antipode map of S , the connected sum of two copies of 3-dimensional projective space, and the product of S with two-dimensional projective space. The first two are mapping tori of the identity map and antipode map of

2522-561: 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

2619-476: 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 . Examples include the 3-sphere , the Poincaré homology sphere , Lens spaces . 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

2716-422: Is a discrete 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

2813-476: Is a diffeomorphism f {\displaystyle f} from M {\displaystyle M} to N {\displaystyle N} . Two C r {\displaystyle C^{r}} -differentiable manifolds are C r {\displaystyle C^{r}} -diffeomorphic if there is an r {\displaystyle r} times continuously differentiable bijective map between them whose inverse

2910-533: Is a homeomorphism, given a pair of manifolds which are diffeomorphic to each other they are in particular homeomorphic to each other. The converse is not true in general. While it is easy to find homeomorphisms that are not diffeomorphisms, it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2 and 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs exist. The first such example

3007-507: 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 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

Geometrization conjecture - Misplaced Pages Continue

3104-417: Is also r {\displaystyle r} times continuously differentiable. Given a subset X {\displaystyle X} of a manifold M {\displaystyle M} and a subset Y {\displaystyle Y} of a manifold N {\displaystyle N} , a function f : X → Y {\displaystyle f:X\to Y}

3201-512: Is always metrizable . When the manifold is not compact, the strong topology captures the behavior of functions "at infinity" and is not metrizable. It is, however, still Baire . Fixing a Riemannian metric on M {\displaystyle M} , the weak topology is the topology induced by the family of metrics as K {\displaystyle K} varies over compact subsets of M {\displaystyle M} . Indeed, since M {\displaystyle M}

3298-411: Is an algebraic limit of geometrically finite Kleinian groups, and was independently proven by Ohshika and Namazi–Souto in 2011 and 2012 respectively. In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. Such structures had been studied prior to Thurston, but his work, particularly the next theorem, would bring them to prominence. In 1981, he announced

3395-683: Is an extension of f {\displaystyle f} ). The function f {\displaystyle f} is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth. Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem: If U {\displaystyle U} , V {\displaystyle V} are connected open subsets of R n {\displaystyle \mathbb {R} ^{n}} such that V {\displaystyle V}

3492-399: Is bijective at each point, f {\displaystyle f} is not invertible because it fails to be injective (e.g. f ( 1 , 0 ) = ( 1 , 0 ) = f ( − 1 , 0 ) {\displaystyle f(1,0)=(1,0)=f(-1,0)} ). Since the differential at a point (for a differentiable function) is a linear map , it has

3589-500: Is differentiable as well. If these functions are r {\displaystyle r} times continuously differentiable, f {\displaystyle f} is called a C r {\displaystyle C^{r}} -diffeomorphism. Two manifolds M {\displaystyle M} and N {\displaystyle N} are diffeomorphic (usually denoted M ≃ N {\displaystyle M\simeq N} ) if there

3686-496: Is essential for V {\displaystyle V} to be simply connected for the function f {\displaystyle f} to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of the complex square function Then f {\displaystyle f} is surjective and it satisfies Thus, though D f x {\displaystyle Df_{x}}

3783-478: Is given in the article on Seifert fiber spaces . Under Ricci flow, manifolds with Euclidean geometry remain invariant. 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 . Other examples are given by

3880-459: Is injective, f {\displaystyle f} is said to be an immersion (or, locally, a "local immersion"). A differentiable bijection is not necessarily a diffeomorphism. f ( x ) = x 3 {\displaystyle f(x)=x^{3}} , for example, is not a diffeomorphism from R {\displaystyle \mathbb {R} } to itself because its derivative vanishes at 0 (and hence its inverse

3977-464: Is not differentiable at 0). This is an example of a homeomorphism that is not a diffeomorphism. When f {\displaystyle f} is a map between differentiable manifolds, a diffeomorphic f {\displaystyle f} is a stronger condition than a homeomorphic f {\displaystyle f} . For a diffeomorphism, f {\displaystyle f} and its inverse need to be differentiable ; for

Geometrization conjecture - Misplaced Pages Continue

4074-473: 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

4171-529: Is required that for p {\displaystyle p} in U {\displaystyle U} , there is a neighborhood of p {\displaystyle p} in which the Jacobian D f {\displaystyle Df} stays non-singular . Suppose that in a chart of the surface, f ( x , y ) = ( u , v ) . {\displaystyle f(x,y)=(u,v).} The total differential of u

4268-622: Is said to be smooth if for all p {\displaystyle p} in X {\displaystyle X} there is a neighborhood U ⊂ M {\displaystyle U\subset M} of p {\displaystyle p} and a smooth function g : U → N {\displaystyle g:U\to N} such that the restrictions agree: g | U ∩ X = f | U ∩ X {\displaystyle g_{|U\cap X}=f_{|U\cap X}} (note that g {\displaystyle g}

4365-455: 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 with quotient R , where R acts on R with 2 (real) eigenspaces, with distinct real eigenvalues of product 1. This is the Bianchi group of type VI 0 and

4462-419: Is the most prevalent geometry in this picture and also the most complicated. The conjecture was proved by Grigori Perelman in 2002–2003. Thurston and Dennis Sullivan generalized Lipman Bers ' density conjecture from singly degenerate Kleinian surface groups to all finitely generated Kleinian groups in the late 1970s and early 1980s. The conjecture states that every finitely generated Kleinian group

4559-422: Is understood, Diff ( M ) {\displaystyle {\text{Diff}}(M)} . This is a "large" group, in the sense that—provided M {\displaystyle M} is not zero-dimensional—it is not locally compact . The diffeomorphism group has two natural topologies : weak and strong ( Hirsch 1997 ). When the manifold is compact , these two topologies agree. The weak topology

4656-788: The European Mathematical Society . In four dimensions, only a rather restricted class of closed 4-manifolds admit a geometric decomposition. However, lists of maximal model geometries can still be given. The four-dimensional maximal model geometries were classified by Richard Filipkiewicz in 1983. They number eighteen, plus one countably infinite family: their usual names are E , Nil, Nil × E , Sol m , n (a countably infinite family), Sol 0 , Sol 1 , H × E , S L ~ {\displaystyle {\widetilde {\rm {SL}}}} × E , H × E , H × H , H , H ( C ) (a complex hyperbolic space ), F (the tangent bundle of

4753-490: The JSJ decomposition , 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 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

4850-469: The Lie bracket of vector fields . Somewhat formally, this is seen by making a small change to the coordinate x {\displaystyle x} at each point in space: so the infinitesimal generators are the vector fields For a connected manifold M {\displaystyle M} , the diffeomorphism group acts transitively on M {\displaystyle M} . More generally,

4947-685: The Oswald Veblen Prize in Geometry . Thurston received the Fields Medal in 1982 for "revolutioniz[ing] [the] study of topology in 2 and 3 dimensions, showing interplay between analysis, topology, and geometry" and "contribut[ing] [the] idea that a very large class of closed 3-manifolds carry a hyperbolic structure." In 2005, Thurston won the first American Mathematical Society Book Prize , for Three-dimensional Geometry and Topology . The prize "recognizes an outstanding research book that makes

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5044-426: The Poincaré conjecture and Thurston's elliptization conjecture . Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print. Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at

5141-516: The Seifert–Weber space , or "sufficiently complicated" Dehn surgeries on links , or most Haken manifolds . The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible, atoroidal , and has infinite fundamental group. This geometry can be modeled as a left invariant metric on the Bianchi group of type V or VII h≠0 . Under Ricci flow, manifolds with hyperbolic geometry expand. The point stabilizer

5238-522: The SnapPea computing program. During Thurston's directorship at MSRI, the institute introduced several innovative educational programs that have since become standard for research institutes. His Ph.D. students include Danny Calegari , Richard Canary , David Gabai , William Goldman , Benson Farb , Richard Kenyon , Steven Kerckhoff , Yair Minsky , Igor Rivin , Oded Schramm , Richard Schwartz , William Floyd , and Jeffrey Weeks. His early work, in

5335-442: The figure-eight knot complement was hyperbolic . This was the first example of a hyperbolic knot . Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure-eight knot complement. He showed that the figure-eight knot complement could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave

5432-466: The mapping torus of a finite-order automorphism of the 2-torus; see torus bundle . 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 VII 0 . Finite volume manifolds with this geometry are all compact, and have the structure of a Seifert fiber space (sometimes in two ways). The complete list of such manifolds

5529-515: The orbifold theorem , an extension of his geometrization theorem to the setting of 3-orbifolds. Two teams of mathematicians around 2000 finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton. His original proof relied partly on Richard S. Hamilton 's work on the Ricci flow . In 1976, Thurston and James Harris Simons shared

5626-597: The outer automorphism group of the fundamental group of the surface. William Thurston refined this analysis by classifying elements of the mapping class group into three types: those equivalent to a periodic diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent to pseudo-Anosov diffeomorphisms . In the case of the torus S 1 × S 1 = R 2 / Z 2 {\displaystyle S^{1}\times S^{1}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}} ,

5723-431: The spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture. A 3-manifold is called closed if it is compact and has no boundary . Every closed 3-manifold has a prime decomposition : this means it is the connected sum of prime 3-manifolds (this decomposition is essentially unique except for

5820-496: 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 (often in several ways). Under normalized Ricci flow manifolds with this geometry converge to

5917-641: The 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries . (There are also uncountably 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 S × 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. The point stabilizer

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6014-439: The appropriate complex number plane. As such, there is a type of angle ( Euclidean , hyperbolic , or slope ) that is preserved in such a multiplication. Due to Df being invertible, the type of complex number is uniform over the surface. Consequently, a surface deformation or diffeomorphism of surfaces has the conformal property of preserving (the appropriate type of) angles. Let M {\displaystyle M} be

6111-606: 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 for his work, and in 2010 the Clay Mathematics Institute awarded him its 1 million USD prize for solving the Poincare conjecture, though Perelman declined both awards. The Poincaré conjecture and

6208-476: 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 . 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,

6305-463: The ball B n {\displaystyle B^{n}} . For manifolds, the diffeomorphism group is usually not connected. Its component group is called the mapping class group . In dimension 2 (i.e. surfaces ), the mapping class group is a finitely presented group generated by Dehn twists ; this has been proved by Max Dehn , W. B. R. Lickorish , and Allen Hatcher ). Max Dehn and Jakob Nielsen showed that it can be identified with

6402-481: 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

6499-410: The choice of initial metric. The Fields Medal was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for Haken manifolds . In 1982, Richard S. Hamilton showed that given a closed 3-manifold with a metric of positive Ricci curvature , the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as

6596-409: The circle is pathwise connected. This can be seen by noting that any such diffeomorphism can be lifted to a diffeomorphism f {\displaystyle f} of the reals satisfying [ f ( x + 1 ) = f ( x ) + 1 ] {\displaystyle [f(x+1)=f(x)+1]} ; this space is convex and hence path-connected. A smooth, eventually constant path to

6693-415: The diffeomorphism group acts transitively on the configuration space C k M {\displaystyle C_{k}M} . If M {\displaystyle M} is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space F k M {\displaystyle F_{k}M} and the action on M {\displaystyle M}

6790-408: The diffeomorphism group into a Banach manifold with smooth right translations; left translations and inversion are only continuous. If r = ∞ {\displaystyle r=\infty } , the space of vector fields is a Fréchet space . Moreover, the transition maps are smooth, making the diffeomorphism group into a Fréchet manifold and even into a regular Fréchet Lie group . If

6887-437: 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: It

6984-516: The early 1970s, was mainly in foliation theory. His more significant results include: In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that it led to an exodus from the field, where advisors counselled students against going into foliation theory, because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6 ). His later work, starting around

7081-459: The first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken. These examples were actually hyperbolic and motivated his next theorem. Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem. To complete

7178-510: The function and its inverse are continuously differentiable . Given two differentiable manifolds M {\displaystyle M} and N {\displaystyle N} , a differentiable map f : M → N {\displaystyle f\colon M\rightarrow N} is a diffeomorphism if it is a bijection and its inverse f − 1 : N → M {\displaystyle f^{-1}\colon N\rightarrow M}

7275-613: 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 of an Anosov map 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 ( 2 1 1 1 ) {\displaystyle \left({\begin{array}{*{20}c}2&1\\1&1\\\end{array}}\right)} ), or quotients of these by groups of order at most 8. The eigenvalues of

7372-643: The hyperbolic plane), S × E , S × H , S × E , S , CP (the complex projective plane ), and S × S . No closed manifold admits the geometry F , but there are manifolds with proper decomposition including an F piece. The five-dimensional maximal model geometries were classified by Andrew Geng in 2016. There are 53 individual geometries and six infinite families. Some new phenomena not observed in lower dimensions occur, including two uncountable families of geometries and geometries with no compact quotients. William Thurston William Paul Thurston (October 30, 1946 – August 21, 2012)

7469-409: The hyperbolic structure on the figure-eight knot complement. By utilizing Haken 's normal surface techniques, he classified the incompressible surfaces in the knot complement. Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 Dehn surgeries on the figure-eight knot resulted in irreducible , non- Haken non- Seifert-fibered 3-manifolds. These were

7566-534: The identity gives a second more elementary way of extending a diffeomorphism from the circle to the open unit disc (a special case of the Alexander trick ). Moreover, the diffeomorphism group of the circle has the homotopy-type of the orthogonal group O ( 2 ) {\displaystyle O(2)} . The corresponding extension problem for diffeomorphisms of higher-dimensional spheres S n − 1 {\displaystyle S^{n-1}}

7663-838: The images of ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } . The map ψ f ϕ − 1 : U → V {\displaystyle \psi f\phi ^{-1}:U\to V} is then a diffeomorphism as in the definition above, whenever f ( ϕ − 1 ( U ) ) ⊆ ψ − 1 ( V ) {\displaystyle f(\phi ^{-1}(U))\subseteq \psi ^{-1}(V)} . Since any manifold can be locally parametrised, we can consider some explicit maps from R 2 {\displaystyle \mathbb {R} ^{2}} into R 2 {\displaystyle \mathbb {R} ^{2}} . In mechanics ,

7760-560: The manifold is σ {\displaystyle \sigma } -compact and not compact the full diffeomorphism group is not locally contractible for any of the two topologies. One has to restrict the group by controlling the deviation from the identity near infinity to obtain a diffeomorphism group which is a manifold; see ( Michor & Mumford 2013 ). The Lie algebra of the diffeomorphism group of M {\displaystyle M} consists of all vector fields on M {\displaystyle M} equipped with

7857-433: The mapping class group is simply the modular group SL ( 2 , Z ) {\displaystyle {\text{SL}}(2,\mathbb {Z} )} and the classification becomes classical in terms of elliptic , parabolic and hyperbolic matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on a compactification of Teichmüller space ; as this enlarged space

7954-486: The metric becomes "almost round" just before the collapse. He later developed a program to prove the geometrization conjecture by Ricci flow with surgery . 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

8051-512: The mid-1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert–Weber space . The independent and distinct approaches of Robert Riley and Troels Jørgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that

8148-491: 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 spaces . 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

8245-427: The picture, Thurston proved a hyperbolization theorem for Haken manifolds . A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance. The hyperbolization theorem for Haken manifolds has been called Thurston's Monster Theorem, due to the length and difficulty of

8342-406: The pieces of the manifold with the "positive curvature" geometries S and S × 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 2003, Grigori Perelman announced a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has

8439-497: The proof. Complete proofs were not written up until almost 20 years later. The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds . Thurston was next led to formulate his geometrization conjecture . This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric decomposition involving eight geometries, now called Thurston model geometries. Hyperbolic geometry

8536-427: The same for N {\displaystyle N} . Let ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } be charts on, respectively, M {\displaystyle M} and N {\displaystyle N} , with U {\displaystyle U} and V {\displaystyle V} as, respectively,

8633-405: The structure of a Seifert fiber space (often in several ways). The complete list of such manifolds is given in the article on spherical 3-manifolds . Under Ricci flow, manifolds with this geometry collapse to a point in finite time. The point stabilizer is O(3, R ), and the group G is the 6-dimensional Lie group R × O(3, R ), with 2 components. Examples are the 3-torus , and more generally

8730-465: The structure of a Seifert fiber space . The classification of such manifolds is given in the article on Seifert fiber spaces . Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold. This fibers over E , and so is sometimes known as "Twisted E × R". It is the geometry of the Heisenberg group . The point stabilizer is O(2, R ). The group G has 2 components, and

8827-408: The structure of a Seifert fiber space . The classification of such manifolds is given in the article on Seifert fiber spaces . Under normalized Ricci flow, compact manifolds with this geometry converge to R with the flat metric. 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

8924-411: 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 . Thurston was a professor of mathematics at Princeton University , University of California, Davis , and Cornell University . He was also a director of the Mathematical Sciences Research Institute . William Thurston

9021-612: Was born in Washington, D.C. , to Margaret Thurston ( née  Martt ), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital strabismus as a child, causing issues with depth perception. His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones. He received his bachelor's degree from New College in 1967 as part of its inaugural class. For his undergraduate thesis, he developed an intuitionist foundation for topology. Following this, he received

9118-457: Was constructed by John Milnor in dimension 7. He constructed a smooth 7-dimensional manifold (called now Milnor's sphere ) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are, in fact, 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is the total space of a fiber bundle over the 4-sphere with the 3-sphere as the fiber). More unusual phenomena occur for 4-manifolds . In

9215-491: Was homeomorphic to a closed ball, the Brouwer fixed-point theorem became applicable. Smale conjectured that if M {\displaystyle M} is an oriented smooth closed manifold, the identity component of the group of orientation-preserving diffeomorphisms is simple . This had first been proved for a product of circles by Michel Herman ; it was proved in full generality by Thurston. Since every diffeomorphism

9312-451: Was much studied in the 1950s and 1960s, with notable contributions from René Thom , John Milnor and Stephen Smale . An obstruction to such extensions is given by the finite abelian group Γ n {\displaystyle \Gamma _{n}} , the " group of twisted spheres ", defined as the quotient of the abelian component group of the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of

9409-400: 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 Laurent Bessières and co-authors, which uses Thurston's hyperbolization theorem for Haken manifolds and Gromov 's norm for 3-manifolds. A book by the same authors with complete details of their version of the proof has been published by

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