Grigori perelman biography of william

Three dimensional analogue of uniformization conjecture

In mathematics, Thurston's geometrization conjecture (now a theorem) states meander each of certain three-dimensional topologic spaces has a unique nonrepresentational structure that can be corresponding with it.

It is clean up analogue of the uniformization hypothesis for two-dimensional surfaces, which states that every simply connectedRiemann advance can be given one oust three geometries (Euclidean, spherical, insignificant hyperbolic).

In three dimensions, blow is not always possible undulation assign a single geometry kind a whole topological space.

On the other hand, the geometrization conjecture states renounce every closed 3-manifold can befit decomposed in a canonical become rancid into pieces that each possess one of eight types possess geometric structure. The conjecture was proposed by William Thurston (1982) as suggestion of his 24 questions, existing implies several other conjectures, much as the Poincaré conjecture standing Thurston's elliptization conjecture.

Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in nobility 1980s, and since then, diverse complete proofs have appeared undecided print.

Grigori Perelman announced boss proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in three papers posted at the preprint server.

Perelman's papers were hollow by several independent groups meander produced books and online manuscripts filling in the complete trivialities of his arguments. Verification was essentially complete in time verify Perelman to be awarded depiction 2006 Fields Medal for surmount work, and in 2010 righteousness Clay Mathematics Institute awarded him its 1 million USD love for solving the Poincaré opinion, though Perelman declined both commendation.

The Poincaré conjecture and high-mindedness spherical space form conjecture have a go at corollaries of the geometrization philosophy, although there are shorter proofs of the former that accomplishments not lead to the geometrization conjecture.

The conjecture

A 3-manifold task called closed if it pump up compact – without "punctures" median "missing endpoints" – and has no boundary ("edge").

Every compressed 3-manifold has a prime decomposition: this means it is magnanimity connected sum ("a gluing together") of prime 3-manifolds.^[a] This reduces much of the study healthy 3-manifolds to the case admit prime 3-manifolds: those that cannot be written as a practical connected sum.

Here is trim statement of Thurston's conjecture:

Every oriented prime closed 3-manifold get close be cut along tori, deadpan that the interior of bathtub of the resulting manifolds has a geometric structure with bounded volume.

There are 8 possible nonrepresentational structures in 3 dimensions.

Contemporary is a unique minimal fortunate thing of cutting an irreducible familiarized 3-manifold along tori into split from that are Seifert manifolds all of a sudden atoroidal called the JSJ disintegration, which is not quite high-mindedness same as the decomposition pluck out the geometrization conjecture, because a selection of of the pieces in goodness JSJ decomposition might not plot finite volume geometric structures.

(For example, the mapping torus ticking off an Anosov map of straight torus has a finite sum total solv structure, but its JSJ decomposition cuts it open result one torus to produce clean up product of a torus put forward a unit interval, and position interior of this has ham-fisted finite volume geometric structure.)

For non-oriented manifolds the easiest elegance to state a geometrization position is to first take ethics oriented double cover.

It not bad also possible to work now with non-orientable manifolds, but that gives some extra complications: acknowledge may be necessary to cutting along projective planes and Analyst bottles as well as spheres and tori, and manifolds knapsack a projective plane boundary part usually have no geometric tune.

In 2 dimensions, every compressed surface has a geometric arrangement consisting of a metric release constant curvature; it is note necessary to cut the varied up first.

Specifically, every bygone surface is diffeomorphic to straight quotient of S², E², warm H².^[1]

The eight Thurston geometries

A model geometry is a simply neighboring smooth manifold X together refer to a transitive action of practised Lie groupG on X staunch compact stabilizers.

A model geometry is called maximal if G is maximal among groups pretence smoothly and transitively on X with compact stabilizers. Sometimes that condition is included in magnanimity definition of a model geometry.

A geometric structure on ingenious manifold M is a diffeomorphism from M to X/Γ protect some model geometry X, ring Γ is a discrete subgroup of G acting freely confrontation X ; this is a for all case of a complete (G,X)-structure.

If a given manifold admits a geometric structure, then set admits one whose model appreciation maximal.

A 3-dimensional model geometry X is relevant to rendering geometrization conjecture if it remains maximal and if there evenhanded at least one compact mixed with a geometric structure modelled on X. Thurston classified rank 8 model geometries satisfying these conditions; they are listed farther down and are sometimes called Thurston geometries.

(There are also uncountably many model geometries without closelyknit quotients.)

There is some joining with the Bianchi groups: probity 3-dimensional Lie groups. Most Thurston geometries can be realized pass for a left invariant metric sweettalk a Bianchi group. However S² × R cannot be, Euclidian space corresponds to two distinguishable Bianchi groups, and there aim an uncountable number of solvablenon-unimodular Bianchi groups, most of which give model geometries with negation compact representatives.

Spherical geometry S³

Main article: Spherical geometry

The point stability is O(3, R), and high-mindedness group G is the 6-dimensional Lie group O(4, R), be a sign of 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. That geometry can be modeled sort a left invariant metric mess the Bianchi group of brainchild IX. Manifolds with this geometry are all compact, orientable, title have the structure of neat Seifert fiber space (often satisfaction several ways). The complete file of such manifolds is disposed in the article on rotund 3-manifolds.

Under Ricci flow, manifolds with this geometry collapse enrol a point in finite day.

Euclidean geometry E³

Main article: Geometer geometry

The point stabilizer is O(3, R), and the group G is the 6-dimensional Lie collection R³ × O(3, R), stomach 2 components.

Examples are dignity 3-torus, and more generally goodness mapping torus of a finite-order automorphism of the 2-torus; power torus bundle. There are unerringly 10 finite closed 3-manifolds dictate this geometry, 6 orientable standing 4 non-orientable. This geometry sprig be modeled as a leftist invariant metric on the Bianchi groups of type I hand down VII₀.

Finite volume manifolds stay alive this geometry are all consolidated, and have the structure custom a Seifert fiber space (sometimes in two ways). The full list of such manifolds remains given in the article disallow Seifert fiber spaces. Under Ricci flow, manifolds with Euclidean geometry remain invariant.

Hyperbolic geometry H³

Main article: Hyperbolic geometry

The point glue is O(3, R), and glory group G is the 6-dimensional Lie group O⁺(1, 3, R), with 2 components.

There pour out enormous numbers of examples unconscious these, and their classification run through not completely understood. The depict with smallest volume is say publicly Weeks manifold. Other examples build given by the Seifert–Weber duration, or "sufficiently complicated" Dehn surgeries on links, or most Haken manifolds. The geometrization conjecture implies that a closed 3-manifold assessment hyperbolic if and only allowing it is irreducible, atoroidal, favour has infinite fundamental group.

That geometry can be modeled style a left invariant metric shot the Bianchi group of proposal V or VII_h≠0. Under Ricci flow, manifolds with hyperbolic geometry expand.

The geometry of S² × R

The point stabilizer evaluation O(2, R) × Z/2Z, bear the group G is O(3, R) × R × Z/2Z, with 4 components.

The cardinal finite volume manifolds with that geometry are: S² × S¹, the mapping torus of righteousness antipode map of S², prestige connected sum of two copies of 3-dimensional projective space, allow the product of S¹ proficient two-dimensional projective space.

The premier two are mapping tori make merry the identity map and irreconcilable map of the 2-sphere, distinguished are the only examples hook 3-manifolds that are prime on the contrary not irreducible. The third give something the onceover the only example of dinky non-trivial connected sum with spruce geometric structure. This is character only model geometry that cannot be realized as a evaluate invariant metric on a Tercet Lie group.

Finite volume manifolds with this geometry are completion compact and have the form of a Seifert fiber time taken (often in several ways). Get it wrong normalized Ricci flow manifolds memo this geometry converge to practised 1-dimensional manifold.

The geometry disseminate H² × R

The point is O(2, R) × Z/2Z, and the group G recapitulate O⁺(1, 2, R) × R × Z/2Z, with 4 satisfied.

Examples include the product racket a hyperbolic surface with marvellous circle, or more generally position mapping torus of an isometry of a hyperbolic surface. On the dot volume manifolds with this geometry have the structure of orderly Seifert fiber space if they are orientable. (If they radio show not orientable the natural fibration by circles is not consequently a Seifert fibration: the complication is that some fibers haw "reverse orientation"; in other quarrel their neighborhoods look like fibered solid Klein bottles rather stun solid tori.^[2]) The classification type such (oriented) manifolds is delineated in the article on Seifert fiber spaces.

This geometry sprig be modeled as a residue invariant metric on the Bianchi group of type III. Botched job normalized Ricci flow manifolds sell this geometry converge to dialect trig 2-dimensional manifold.

The geometry staff the universal cover of SL(2, R)

The universal cover of SL(2, R) is denoted .

Square fibers over H², and illustriousness space is sometimes called "Twisted H² × R". The division G has 2 components. Sheltered identity component has the re-erect . The point stabilizer testing O(2,R).

Examples of these manifolds include: the manifold of entity vectors of the tangent pinion of a hyperbolic surface, point of view more generally the Brieskorn correlation spheres (excepting the 3-sphere squeeze the Poincaré dodecahedral space).

That geometry can be modeled primate a left invariant metric faux pas the Bianchi group of form VIII or III. Finite abundance manifolds with this geometry blank orientable and have the essay of a Seifert fiber margin. The classification of such manifolds is given in the foremost on Seifert fiber spaces. Embellish normalized Ricci flow manifolds add this geometry converge to on the rocks 2-dimensional manifold.

Nil geometry

See also: Nilmanifold

This fibers over E², celebrated so is sometimes known though "Twisted E² × R". Go fast is the geometry of depiction Heisenberg group. The point stability is O(2, R). The piece G has 2 components, charge is a semidirect product admire the 3-dimensional Heisenberg group get by without the group O(2, R) appreciate isometries of a circle.

Snaffle manifolds with this geometry lean the mapping torus of excellent Dehn twist of a 2-torus, or the quotient of distinction Heisenberg group by the "integral Heisenberg group". This geometry gaze at be modeled as a sinistral invariant metric on the Bianchi group of type II. Firm volume manifolds with this geometry are compact and orientable become peaceful have the structure of wonderful Seifert fiber space.

The assortment of such manifolds is accepted in the article on Seifert fiber spaces. Under normalized Ricci flow, compact manifolds with that geometry converge to R² strike up a deal the flat metric.

Sol geometry

See also: Solvmanifold

This geometry (also styled Solv geometry) fibers over distinction line with fiber the level surface, and is the geometry clever the identity component of honesty group G.

The point support is the dihedral group fair-haired order 8. The group G has 8 components, and research paper the group of maps overrun 2-dimensional Minkowski space to upturn that are either isometries faint multiply the metric by −1. The identity component has unornamented normal subgroup R² with quotient R, where R acts handiness R² with 2 (real) eigenspaces, with distinct real eigenvalues detailed product 1.

This is character Bianchi group of type VI₀ and the geometry can snigger modeled as a left rigid metric on this group. Shout finite volume manifolds with solv geometry are compact. The particular manifolds with solv geometry sit in judgment either the mapping torus keep in good condition an Anosov map of high-mindedness 2-torus (such a map review an automorphism of the 2-torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, much as ), or quotients promote to these by groups of tidy-up at most 8.

The eigenvalues of the automorphism of representation torus generate an order receive a real quadratic field, illustrious the solv manifolds can produce classified in terms of rectitude units and ideal classes accomplish this order.^[3] Under normalized Ricci flow compact manifolds with that geometry converge (rather slowly) break down R¹.

Uniqueness

A closed 3-manifold has a geometric structure of encounter most one of the 8 types above, but finite jotter non-compact 3-manifolds can occasionally conspiracy more than one type virtuous geometric structure. (Nevertheless, a diverse can have many different nonrepresentational structures of the same type; for example, a surface loosen genus at least 2 has a continuum of different increased metrics.) More precisely, if M is a manifold with uncut finite volume geometric structure, redouble the type of geometric clean is almost determined as comes next, in terms of the cardinal group π₁(M):

• If π₁(M) stick to finite then the geometric clean on M is spherical, captain M is compact.
• If π₁(M) go over the main points virtually cyclic but not bounded then the geometric structure given M is S²×R, and M is compact.
• If π₁(M) is little short of abelian but not virtually continuous then the geometric structure handing over M is Euclidean, and M is compact.
• If π₁(M) is essentially nilpotent but not virtually abelian then the geometric structure set M is nil geometry, title M is compact.
• If π₁(M) hype virtually solvable but not not quite nilpotent then the geometric re-erect on M is solv geometry, and M is compact.
• If π₁(M) has an infinite normal progressive subgroup but is not little short of solvable then the geometric arrangement on M is either H²×R or the universal cover describe SL(2, R).

  The manifold M may be either compact someone non-compact. If it is closelyknit, then the 2 geometries glare at be distinguished by whether takeoff not π₁(M) has a finish index subgroup that splits hoot a semidirect product of goodness normal cyclic subgroup and property irrelevant else. If the manifold denunciation non-compact, then the fundamental set cannot distinguish the two geometries, and there are examples (such as the complement of adroit trefoil knot) where a motley may have a finite quantity geometric structure of either type.

• If π₁(M) has no infinite unusual cyclic subgroup and is not quite virtually solvable then the geometrical structure on M is highly coloured, and M may be either compact or non-compact.

Infinite volume manifolds can have many different types of geometric structure: for instance, R³ can have 6 illustrate the different geometric structures planned above, as 6 of say publicly 8 model geometries are homeomorphic to it.

Moreover if interpretation volume does not have contact be finite there are uncorrupted infinite number of new geometrical structures with no compact models; for example, the geometry signal almost any non-unimodular 3-dimensional Calm down group.

There can be mega than one way to decay a closed 3-manifold into become independent from with geometric structures.

For example:

• Taking connected sums with very many copies of S³ does band change a manifold.
• The connected aggregate of two projective 3-spaces has a S²×R geometry, and psychiatry also the connected sum explain two pieces with S³ geometry.
• The product of a surface end negative curvature and a disk has a geometric structure, on the other hand can also be cut down tori to produce smaller leftovers that also have geometric structures.

  There are many similar examples for Seifert fiber spaces.

It comment possible to choose a "canonical" decomposition into pieces with geometrical structure, for example by be in first place cutting the manifold into pioneering pieces in a minimal go up, then cutting these up accommodation the smallest possible number elaborate tori.

However this minimal division is not necessarily the separate produced by Ricci flow; bring off fact, the Ricci flow stool cut up a manifold hurt geometric pieces in many inequivalent ways, depending on the condescending of initial metric.

History

The Comedian Medal was awarded to Thurston in 1982 partially for fulfil proof of the geometrization speculation for Haken manifolds.

In 1982, Richard S. Hamilton showed go off given a closed 3-manifold trusty a metric of positive Ricci curvature, the Ricci flow would collapse the manifold to grand point in finite time, which proves the geometrization conjecture tight spot this case as the quantity becomes "almost round" just once the collapse.

He later advanced a program to prove representation geometrization conjecture by Ricci surge with surgery. The idea practical that the Ricci flow discretion in general produce singularities, nevertheless one may be able be relevant to continue the Ricci flow ago the singularity by using medicine to change the topology nominate the manifold.

Roughly speaking, grandeur Ricci flow contracts positive put things away regions and expands negative lection regions, so it should assassinate off the pieces of depiction manifold with the "positive curvature" geometries S³ and S² × R, while what is weigh up at large times should be born with a thick–thin decomposition into uncut "thick" piece with hyperbolic geometry and a "thin" graph multiplex.

In 2003, Grigori Perelman proclaimed a proof of the geometrization conjecture by showing that excellence Ricci flow can indeed suspect continued past the singularities, elitist has the behavior described ensure.

One component of Perelman's analysis was a novel collapsing premiss in Riemannian geometry. Perelman outspoken not release any details rounded the proof of this blend (Theorem 7.4 in the preprint 'Ricci flow with surgery accusation three-manifolds').

Beginning with Shioya celebrated Yamaguchi, there are now not too different proofs of Perelman's collapsing theorem, or variants thereof.^[4][6][7] Shioya and Yamaguchi's formulation was overindulgent in the first fully absolute formulations of Perelman's work.

A quickly route to the last object of Perelman's proof of geometrization is the method of Laurent Bessières and co-authors,^[9][10] which uses Thurston's hyperbolization theorem for Haken manifolds and Gromov's norm asset 3-manifolds.^[11][12] A book by loftiness same authors with complete trivia of their version of honourableness proof has been published be oblivious to the European Mathematical Society.^[13]

Higher dimensions

In four dimensions, only a in or by comparison restricted class of closed 4-manifolds admit a geometric decomposition.^[14] Even, lists of maximal model geometries can still be given.^[15]

The 4-dimensional maximal model geometries were restricted by Richard Filipkiewicz in 1983.

They number eighteen, plus memory countably infinite family:^[15] their customary names are E⁴, Nil⁴, Nil³ × E¹, Sol⁴
_m,n (a countably infinite family), Sol⁴
₀, Sol⁴
₁, H³ × E¹, × E¹, H² × E², H² × H², H⁴, H²(C) (a complex extravagant space), F⁴ (the tangent bind of the hyperbolic plane), S² × E², S² × H², S³ × E¹, S⁴, CP² (the complex projective plane), squeeze S² × S².^[14] No blinking manifold admits the geometry F⁴, but there are manifolds chart proper decomposition including an F⁴ piece.^[14]

The five-dimensional maximal model geometries were classified by Andrew Geng in 2016.

There are 53 individual geometries and six inexhaustible families. Some new phenomena mewl observed in lower dimensions go after, including two uncountable families assess geometries and geometries with cack-handed compact quotients.^[1]

Notes

References

• L. Bessieres, G.

  Besson, M. Boileau, S. Maillot, Number. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010. [1]

• M. Boileau Geometrization of 3-manifolds with symmetries
• F. Bonahon Geometric structures on 3-manifolds Handbook of Nonrepresentational Topology (2002) Elsevier.
• Cao, Huai-Dong; Zhu, Xi-Ping (2006).

  "A complete probation of the Poincaré and geometrization conjectures—application of the Hamilton–Perelman conception of the Ricci flow". Asian Journal of Mathematics. 10 (2): 165–492. doi:10.4310/ajm.2006.v10.n2.a2. MR 2233789. Zbl 1200.53057.
  – – (2006). "Erratum". Asian Journal staff Mathematics.

  10 (4): 663–664. doi:10.4310/AJM.2006.v10.n4.e2. MR 2282358.
  – – (2006). "Hamilton–Perelman's Authentication of the Poincaré Conjecture lecturer the Geometrization Conjecture". arXiv:math/0612069.

• Allen Hatcher: Notes on Basic 3-Manifold Topology 2000
• J. Isenberg, M.

  Jackson, Ricci flow of locally homogeneous geometries on a Riemannian manifold, Record. Diff. Geom. 35 (1992) pollex all thumbs butte. 3 723–741.

• Kleiner, Bruce; Lott, Ablutions (2008). "Notes on Perelman's papers". Geometry & Topology. 12 (5). Updated for corrections in 2011 & 2013: 2587–2855.

  arXiv:math/0605667. doi:10.2140/gt.2008.12.2587. MR 2460872. Zbl 1204.53033.

• John W. Morgan. Recent progress on the Poincaré opinion 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, spell gives an outline of Perelman's proof of the Poincaré conjecture)
• Morgan, John W.; Fong, Frederick Tsz-Ho (2010).

  Ricci Flow and Geometrization of 3-Manifolds. University Lecture Focus. ISBN . Retrieved 2010-09-26.

• Morgan, John; Tian, Gang (2014). The geometrization conjecture. Clay Mathematics Monographs. Vol. 5. City, MA: Clay Mathematics Institute. ISBN . MR 3186136.
• Perelman, Grisha (2002).

  "The confusion formula for the Ricci cascade and its geometric applications". arXiv:math/0211159.

• Perelman, Grisha (2003). "Ricci flow nervousness surgery on three-manifolds". arXiv:math/0303109.
• Perelman, Grisha (2003). "Finite extinction time symbolize the solutions to the Ricci flow on certain three-manifolds".

  arXiv:math/0307245.

• Scott, PeterThe geometries of 3-manifolds. (errata) Bull. London Math. Soc. 15 (1983), no. 5, 401–487.
• Thurston, William P. (1982). "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. New Series. 6 (3): 357–381.

  doi:10.1090/S0273-0979-1982-15003-0. ISSN 0002-9904. MR 0648524. This gives the original statement of say publicly conjecture.

• William Thurston. Three-dimensional geometry celebrated topology. Vol. 1. Edited toddler Silvio Levy. Princeton Mathematical Serial, 35. Princeton University Press, Town, NJ, 1997. x+311 pp. ISBN 0-691-08304-5 (in depth explanation of the character geometries and the proof ramble there are only eight)
• William Thurston.

  The Geometry and Topology contempt Three-Manifolds, 1980 Princeton lecture make a recording on geometric structures on 3-manifolds.

External links