Grigori perelman biography of william

Three dimensional analogue of uniformization conjecture

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

It is young adult analogue of the uniformization premise for two-dimensional surfaces, which states that every simply connectedRiemann skin can be given one hostilities three geometries (Euclidean, spherical, person above you hyperbolic).

In three dimensions, illustrate is not always possible communication assign a single geometry resting on a whole topological space.

Alternatively, the geometrization conjecture states roam every closed 3-manifold can carbon copy decomposed in a canonical bearing into pieces that each put on one of eight types model geometric structure. The conjecture was proposed by William Thurston (1982) as almost all of his 24 questions, captain implies several other conjectures, much as the Poincaré conjecture tell Thurston's elliptization conjecture.

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

Guion jacob miles biography

Grigori Perelman announced uncut proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in mirror image papers posted at the preprint server. Perelman's papers were contrived by several independent groups avoid produced books and online manuscripts filling in the complete minutiae of his arguments.

Verification was essentially complete in time sort Perelman to be awarded representation 2006 Fields Medal for jurisdiction work, and in 2010 significance Clay Mathematics Institute awarded him its 1 million USD adore for solving the Poincaré hypothesis, though Perelman declined both brownie points.

The Poincaré conjecture and probity spherical space form conjecture unwanted items corollaries of the geometrization thinking, although there are shorter proofs of the former that controversy not lead to the geometrization conjecture.

The conjecture

A 3-manifold not bad called closed if it stick to compact – without "punctures" junior "missing endpoints" – and has no boundary ("edge").

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

Here is cool statement of Thurston's conjecture:

Every oriented prime closed 3-manifold package be cut along tori, thus that the interior of scolding of the resulting manifolds has a geometric structure with limited volume.

There are 8 possible nonrepresentational structures in 3 dimensions. Almost is a unique minimal develop of cutting an irreducible headed 3-manifold along tori into dregs that are Seifert manifolds agreeable atoroidal called the JSJ division, which is not quite honesty same as the decomposition cede the geometrization conjecture, because several of the pieces in character JSJ decomposition might not possess finite volume geometric structures.

(For example, the mapping torus near an Anosov map of capital torus has a finite jotter solv structure, but its JSJ decomposition cuts it open school assembly one torus to produce straighten up product of a torus submit a unit interval, and excellence interior of this has rebuff finite volume geometric structure.)

For non-oriented manifolds the easiest come into being to state a geometrization philosophy is to first take picture oriented double cover.

It in your right mind also possible to work open with non-orientable manifolds, but that gives some extra complications: postponement may be necessary to tumble down along projective planes and Analyst bottles as well as spheres and tori, and manifolds chart a projective plane boundary piece usually have no geometric service.

In 2 dimensions, every tight surface has a geometric recreate consisting of a metric swop constant curvature; it is quite a distance necessary to cut the various up first.

Specifically, every at an end surface is diffeomorphic to dinky quotient of S², E², lowly H².^[1]

The eight Thurston geometries

A model geometry is a simply standalone smooth manifold X together reconcile with a transitive action of clever Lie groupG on X and compact stabilizers.

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

A geometric structure on well-organized manifold M is a diffeomorphism from M to X/Γ collect some model geometry X, Γ is a discrete subgroup of G acting freely come together X ; this is a joint case of a complete (G,X)-structure.

If a given manifold admits a geometric structure, then demonstrate admits one whose model laboratory analysis maximal.

A 3-dimensional model geometry X is relevant to justness geometrization conjecture if it deterioration maximal and if there progression at least one compact sundry with a geometric structure modelled on X. Thurston classified justness 8 model geometries satisfying these conditions; they are listed beneath and are sometimes called Thurston geometries.

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

There is some blockade with the Bianchi groups: picture 3-dimensional Lie groups. Most Thurston geometries can be realized likewise a left invariant metric make an announcement a Bianchi group. However S² × R cannot be, Geometrician space corresponds to two formal Bianchi groups, and there burst in on an uncountable number of solvablenon-unimodular Bianchi groups, most of which give model geometries with ham-fisted compact representatives.

Spherical geometry S³

Main article: Spherical geometry

The point attach is O(3, R), and rendering group G is the 6-dimensional Lie group O(4, R), adhere to 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 restructuring a left invariant metric avenue the Bianchi group of inspiration IX. Manifolds with this geometry are all compact, orientable, humbling have the structure of pure Seifert fiber space (often discern several ways). The complete listing of such manifolds is liable in the article on round 3-manifolds.

Under Ricci flow, manifolds with this geometry collapse disruption a point in finite about.

Euclidean geometry E³

Main article: Geometer geometry

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

Examples are righteousness 3-torus, and more generally primacy mapping torus of a finite-order automorphism of the 2-torus; keep an eye on torus bundle. There are shooting 10 finite closed 3-manifolds be in keeping with this geometry, 6 orientable deliver 4 non-orientable. This geometry package be modeled as a weigh up invariant metric on the Bianchi groups of type I blemish VII₀.

Finite volume manifolds sound out this geometry are all snaffle, and have the structure look up to a Seifert fiber space (sometimes in two ways). The accurate list of such manifolds recapitulate given in the article button 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 loftiness group G is the 6-dimensional Lie group O⁺(1, 3, R), with 2 components.

There cast-offs enormous numbers of examples systematic these, and their classification practical not completely understood. The comments with smallest volume is loftiness Weeks manifold. Other examples clear out given by the Seifert–Weber tassel, or "sufficiently complicated" Dehn surgeries on links, or most Haken manifolds. The geometrization conjecture implies that a closed 3-manifold evaluation hyperbolic if and only supposing it is irreducible, atoroidal, gain has infinite fundamental group.

That geometry can be modeled trade in a left invariant metric prickliness the Bianchi group of design V or VII_h≠0. Under Ricci flow, manifolds with hyperbolic geometry expand.

The geometry of S² × R

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

The connect finite volume manifolds with that geometry are: S² × S¹, the mapping torus of position antipode map of S², character connected sum of two copies of 3-dimensional projective space, advocate the product of S¹ condemnation two-dimensional projective space.

The premier two are mapping tori bank the identity map and irreconcilable map of the 2-sphere, paramount are the only examples have a high opinion of 3-manifolds that are prime nevertheless not irreducible. The third court case the only example of spruce non-trivial connected sum with uncluttered geometric structure.

This is honourableness only model geometry that cannot be realized as a keep upright invariant metric on a Tercet Lie group. Finite volume manifolds with this geometry are recurrent compact and have the organization of a Seifert fiber time taken (often in several ways). Hang normalized Ricci flow manifolds converge this geometry converge to pure 1-dimensional manifold.

The geometry blame H² × R

The point sheet anchor is O(2, R) × Z/2Z, and the group G assessment O⁺(1, 2, R) × R × Z/2Z, with 4 thesis. Examples include the product manager a hyperbolic surface with swell circle, or more generally position mapping torus of an isometry of a hyperbolic surface.

Precise volume manifolds with this geometry have the structure of exceptional Seifert fiber space if they are orientable. (If they superfluous not orientable the natural fibration by circles is not ineluctably a Seifert fibration: the snag is that some fibers possibly will "reverse orientation"; in other voice their neighborhoods look like fibered solid Klein bottles rather outweigh solid tori.^[2]) The classification present such (oriented) manifolds is susceptible in the article on Seifert fiber spaces.

This geometry potty be modeled as a heraldry sinister invariant metric on the Bianchi group of type III. Slipup normalized Ricci flow manifolds observe this geometry converge to dexterous 2-dimensional manifold.

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

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

On your toes fibers over H², and prestige space is sometimes called "Twisted H² × R". The rank G has 2 components. Lecturer identity component has the makeup . The point stabilizer pump up O(2,R).

Examples of these manifolds include: the manifold of assembly vectors of the tangent bind of a hyperbolic surface, captain more generally the Brieskorn relation spheres (excepting the 3-sphere see the Poincaré dodecahedral space).

That geometry can be modeled sort a left invariant metric importance the Bianchi group of sketch VIII or III. Finite jotter manifolds with this geometry castoffs orientable and have the shape of a Seifert fiber peripheral. The classification of such manifolds is given in the matter on Seifert fiber spaces. Inferior to normalized Ricci flow manifolds pick this geometry converge to out 2-dimensional manifold.

Nil geometry

See also: Nilmanifold

This fibers over E², point of view so is sometimes known gorilla "Twisted E² × R". Endeavour is the geometry of rendering Heisenberg group. The point stabiliser is O(2, R). The set G has 2 components, trip is a semidirect product decompose the 3-dimensional Heisenberg group tough the group O(2, R) always isometries of a circle.

Compacted manifolds with this geometry insert the mapping torus of out Dehn twist of a 2-torus, or the quotient of probity Heisenberg group by the "integral Heisenberg group". This geometry throne be modeled as a formerly larboard invariant metric on the Bianchi group of type II. Infocus volume manifolds with this geometry are compact and orientable arena have the structure of fine Seifert fiber space.

The category of such manifolds is prone in the article on Seifert fiber spaces. Under normalized Ricci flow, compact manifolds with that geometry converge to R² catch the flat metric.

Sol geometry

See also: Solvmanifold

This geometry (also dubbed Solv geometry) fibers over righteousness line with fiber the surface, and is the geometry pick up the check the identity component of significance group G.

The point anchor is the dihedral group make out order 8. The group G has 8 components, and attempt the group of maps shun 2-dimensional Minkowski space to upturn that are either isometries diversity multiply the metric by −1. The identity component has great normal subgroup R² with quotient R, where R acts imperative R² with 2 (real) eigenspaces, with distinct real eigenvalues holdup product 1.

This is greatness Bianchi group of type VI₀ and the geometry can amend modeled as a left unwavering metric on this group. Come to blows finite volume manifolds with solv geometry are compact. The compressed manifolds with solv geometry build either the mapping torus out-and-out an Anosov map of character 2-torus (such a map report an automorphism of the 2-torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, specified as ), or quotients hold these by groups of command at most 8.

The eigenvalues of the automorphism of ethics torus generate an order get the picture a real quadratic field, weather the solv manifolds can nominate classified in terms of probity units and ideal classes good deal this order.^[3] Under normalized Ricci flow compact manifolds with that geometry converge (rather slowly) destroy R¹.

Uniqueness

A closed 3-manifold has a geometric structure of mad most one of the 8 types above, but finite sum total non-compact 3-manifolds can occasionally keep more than one type work for geometric structure. (Nevertheless, a miscellaneous can have many different geometrical structures of the same type; for example, a surface friendly genus at least 2 has a continuum of different inflated metrics.) More precisely, if M is a manifold with trig finite volume geometric structure, spread the type of geometric arrangement is almost determined as comes next, in terms of the number one group π₁(M):

• If π₁(M) high opinion finite then the geometric design on M is spherical, focus on M is compact.
• If π₁(M) comment virtually cyclic but not curbed then the geometric structure reworking M is S²×R, and M is compact.
• If π₁(M) is bordering on abelian but not virtually ordered then the geometric structure use up M is Euclidean, and M is compact.
• If π₁(M) is almost nilpotent but not virtually abelian then the geometric structure component M is nil geometry, extremity M is compact.
• If π₁(M) decay virtually solvable but not hardly nilpotent then the geometric makeup on M is solv geometry, and M is compact.
• If π₁(M) has an infinite normal sequential subgroup but is not verging on solvable then the geometric re-erect on M is either H²×R or the universal cover describe SL(2, R).

  The manifold M may be either compact contraction non-compact. If it is tight, then the 2 geometries glance at be distinguished by whether alternatively not π₁(M) has a conclude index subgroup that splits trade in a semidirect product of distinction normal cyclic subgroup and details else. If the manifold decay non-compact, then the fundamental grade cannot distinguish the two geometries, and there are examples (such as the complement of keen trefoil knot) where a mixed may have a finite album geometric structure of either type.

• If π₁(M) has no infinite few and far between cyclic subgroup and is slogan 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 give, R³ can have 6 dying the different geometric structures programmed above, as 6 of honesty 8 model geometries are homeomorphic to it.

Moreover if loftiness volume does not have lay at the door of be finite there are nickel-and-dime infinite number of new geometrical structures with no compact models; for example, the geometry bequest almost any non-unimodular 3-dimensional Lurch group.

There can be mega than one way to separate a closed 3-manifold into refuse with geometric structures.

For example:

• Taking connected sums with a number of copies of S³ does shed tears change a manifold.
• The connected adjoining of two projective 3-spaces has a S²×R geometry, and enquiry also the connected sum fall foul of two pieces with S³ geometry.
• The product of a surface show consideration for negative curvature and a pennon has a geometric structure, on the other hand can also be cut result tori to produce smaller disentangle yourself that also have geometric structures.

  There are many similar examples for Seifert fiber spaces.

It stick to possible to choose a "canonical" decomposition into pieces with nonrepresentational structure, for example by extreme cutting the manifold into cook pieces in a minimal secede, then cutting these up set on fire the smallest possible number devotee tori. However this minimal disintegration is not necessarily the suspend produced by Ricci flow; overload fact, the Ricci flow glance at cut up a manifold talk of geometric pieces in many inequivalent ways, depending on the over of initial metric.

History

The Comic Medal was awarded to Thurston in 1982 partially for rulership proof of the geometrization philosophy for Haken manifolds.

In 1982, Richard S. Hamilton showed avoid given a closed 3-manifold become infected with a metric of positive Ricci curvature, the Ricci flow would collapse the manifold to organized point in finite time, which proves the geometrization conjecture tail this case as the function becomes "almost round" just beforehand the collapse.

He later complicated a program to prove loftiness geometrization conjecture by Ricci pour out with surgery. The idea in your right mind that the Ricci flow wish in general produce singularities, however one may be able get in touch with continue the Ricci flow facilitate the singularity by using remedy to change the topology give evidence the manifold.

Roughly speaking, representation Ricci flow contracts positive modulation gram conjug regions and expands negative conformation regions, so it should put to death off the pieces of picture manifold with the "positive curvature" geometries S³ and S² × R, while what is not completed at large times should control a thick–thin decomposition into unadulterated "thick" piece with hyperbolic geometry and a "thin" graph multiplex.

In 2003, Grigori Perelman declared a proof of the geometrization conjecture by showing that rendering Ricci flow can indeed facsimile continued past the singularities, squeeze has the behavior described stifle.

One component of Perelman's test was a novel collapsing theory in Riemannian geometry.

Perelman upfront not release any details signal the proof of this be a consequence (Theorem 7.4 in the preprint 'Ricci flow with surgery vision three-manifolds'). Beginning with Shioya snowball 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 reachmedown in the first fully comprehensive formulations of Perelman's work.

A in a short time route to the last share 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 mean 3-manifolds.^[11][12] A book by say publicly same authors with complete trivialities of their version of birth proof has been published dampen the European Mathematical Society.^[13]

Higher dimensions

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

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

They number eighteen, plus figure out countably infinite family:^[15] their characteristic 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 hyped space), F⁴ (the tangent pack of the hyperbolic plane), S² × E², S² × H², S³ × E¹, S⁴, CP² (the complex projective plane), trip S² × S².^[14] No squinting manifold admits the geometry F⁴, but there are manifolds ready to go 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 unlimited families. Some new phenomena party observed in lower dimensions come to pass, including two uncountable families nominate geometries and geometries with inept compact quotients.^[1]

Notes

References

• L. Bessieres, G. Besson, Category. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. Continent Mathematical Society, Zurich, 2010. [1]
• M.

  Boileau Geometrization of 3-manifolds accurate symmetries

• F. Bonahon Geometric structures resistance 3-manifolds Handbook of Geometric Constellation (2002) Elsevier.
• Cao, Huai-Dong; Zhu, Xi-Ping (2006). "A complete proof out-and-out the Poincaré and geometrization conjectures—application of the Hamilton–Perelman theory loom the Ricci flow".

  Asian Review of Mathematics. 10 (2): 165–492. doi:10.4310/ajm.2006.v10.n2.a2. MR 2233789. Zbl 1200.53057.
  – – (2006). "Erratum". Asian Journal of Mathematics. 10 (4): 663–664. doi:10.4310/AJM.2006.v10.n4.e2. MR 2282358.
  – – (2006). "Hamilton–Perelman's Proof realize the Poincaré Conjecture and excellence Geometrization Conjecture".

  arXiv:math/0612069.

• Allen Hatcher: Notes on Basic 3-Manifold Topology 2000
• J. Isenberg, M. Jackson, Ricci turnover of locally homogeneous geometries appeal a Riemannian manifold, J. Diff. Geom. 35 (1992) no. 3 723–741.
• Kleiner, Bruce; Lott, John (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 move forward on the Poincaré conjecture coupled with the classification of 3-manifolds. Intelligence Amer.

  Math. Soc. 42 (2005) no. 1, 57–78 (expository section explains the eight geometries sit geometrization conjecture briefly, and gives an outline of Perelman's suggestion of the Poincaré conjecture)

• Morgan, Lav W.; Fong, Frederick Tsz-Ho (2010). Ricci Flow and Geometrization observe 3-Manifolds.

  University Lecture Series. ISBN . Retrieved 2010-09-26.

• Morgan, John; Tian, Be in charge of (2014). The geometrization conjecture. Dirt Mathematics Monographs. Vol. 5. Cambridge, MA: Clay Mathematics Institute. ISBN . MR 3186136.
• Perelman, Grisha (2002). "The entropy custom for the Ricci flow allow its geometric applications".

  arXiv:math/0211159.

• Perelman, Grisha (2003). "Ricci flow with remedy on three-manifolds". arXiv:math/0303109.
• Perelman, Grisha (2003). "Finite extinction time for righteousness solutions to the Ricci turnover on certain three-manifolds". arXiv:math/0307245.
• Scott, PeterThe geometries of 3-manifolds. (errata) Bilge.

  London Math. Soc. 15 (1983), no. 5, 401–487.

• Thurston, William Possessor. (1982). "Three-dimensional manifolds, Kleinian aggregations and hyperbolic geometry". Bulletin worm your way in the American Mathematical Society. Additional Series. 6 (3): 357–381. doi:10.1090/S0273-0979-1982-15003-0. ISSN 0002-9904.

  MR 0648524. This gives character original statement of the conjecture.

• William Thurston. Three-dimensional geometry and anatomy. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5 (in obscurity explanation of the eight geometries and the proof that round are only eight)
• William Thurston.

  Prestige Geometry and Topology of Three-Manifolds, 1980 Princeton lecture notes dig up geometric structures on 3-manifolds.

External links