Geometrization conjecture
Field | Geometric topology |
---|---|
Conjectured by | William Thurston |
Conjectured in | 1982 |
First proof by | Grigori Perelman |
First proof in | 2006 |
Consequences | Poincaré conjecture Thurston elliptization conjecture |
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). 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
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 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 to accept either award.
The Poincaré conjecture and 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.
The conjecture
A 3-manifold is called closed if it is compact and has no boundary.
Every closed 3-manifold has a
Here is a statement of Thurston's conjecture:
- Every oriented prime closed 3-manifold 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
For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the
In 2 dimensions, every closed surface has a geometric structure consisting of a
The eight Thurston geometries
A model geometry is a simply connected smooth manifold X together with a
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
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
There is some connection with the
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
Euclidean geometry E3
The point stabilizer is O(3, R), and the group G is the 6-dimensional Lie group R3 × O(3, R), with 2 components. Examples are the
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. Other examples are given by 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 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: S2 × S1, the mapping torus of the antipode map of S2, the connected sum of two copies of 3-dimensional projective space, and the product of S1 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 (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
The geometry of the universal cover of SL(2, R)
The
Examples of these manifolds include: the manifold of unit vectors of the tangent bundle of a hyperbolic surface, and more generally the
Nil geometry
This fibers over E2, and so is sometimes known as "Twisted E2 × R". It is the geometry of the Heisenberg group. 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 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. 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 R2 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 R2 with quotient R, where R acts on R2 with 2 (real) eigenspaces, with distinct real eigenvalues of product 1. This is the
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 π1(M):
- If π1(M) is finite then the geometric structure on M is spherical, and M is compact.
- If π1(M) is virtually cyclic but not finite then the geometric structure on M is S2×R, and M is compact.
- If π1(M) is virtually abelian but not virtually cyclic then the geometric structure on M is Euclidean, and M is compact.
- If π1(M) is virtually nilpotent but not virtually abelian then the geometric structure on M is nil geometry, and M is compact.
- If π1(M) is virtually solvable but not virtually nilpotent then the geometric structure on M is solv geometry, and M is compact.
- If π1(M) has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on M is either H2×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 π1(M) has a finite index 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 π1(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, R3 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 S3 does not change a manifold.
- The connected sum of two projective 3-spaces has a S2×R geometry, and is also the connected sum of two pieces with S3 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 was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for Haken manifolds.
In 1982,
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 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.[4][5][6][7] Shioya and Yamaguchi's formulation was used in the first fully detailed formulations of Perelman's work.[8]
A second route to the last part of Perelman's proof of geometrization is the method of
Higher dimensions
In four dimensions, only a rather restricted class of closed 4-manifolds admit a geometric decomposition.[14] However, lists of maximal model geometries can still be given.[15]
The four-dimensional maximal model geometries were classified by Richard Filipkiewicz in 1983. They number eighteen, plus one countably infinite family:[15] their usual 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 hyperbolic space), F4 (the tangent bundle of the hyperbolic plane), S2 × E2, S2 × H2, S3 × E1, S4, CP2 (the complex projective plane), and S2 × S2.[14] No closed manifold admits the geometry F4, but there are manifolds with 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 infinite families. Some new phenomena not observed in lower dimensions occur, including two uncountable families of geometries and geometries with no compact quotients.[1]
Notes
- ^ arXiv:1605.07545 [math.GT].
- .
- S2CID 55731832.
- S2CID 119481.
- ^ Morgan & Tian 2014.
- ^ Kleiner, Bruce; Lott, John (2014). "Locally collapsed 3-manifolds". Astérisque. 365 (7–99).
- S2CID 514106.
- ^ Cao & Zhu 2006; Kleiner & Lott 2008.
- ].
- S2CID 119436601.
- ISBN 1-57146-067-5.
- Gromov, M.(1983). "Volume and bounded cohomology". Inst. Hautes Études Sci. Publ. Math. (56): 5–99.
- ^ 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
- ^ arXiv:math/0212142.
- ^ a b Filipkiewicz, Richard (1983). Four dimensional geometries (PhD thesis). University of Warwick. Retrieved 31 January 2024.
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. [1]
- M. Boileau Geometrization of 3-manifolds with symmetries
- F. Bonahon Geometric structures on 3-manifolds Handbook of Geometric Topology (2002) Elsevier.
- 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, J. Diff. Geom. 35 (1992) no. 3 723–741.
- Zbl 1204.53033.
- 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)
- Morgan, John W.; Fong, Frederick Tsz-Ho (2010). Ricci Flow and Geometrization of 3-Manifolds. University Lecture Series. ISBN 978-0-8218-4963-7. Retrieved 2010-09-26.
- MR 3186136.
- arXiv:math/0211159.
- arXiv:math/0303109.
- arXiv:math/0307245.
- Scott, Peter The geometries of 3-manifolds. (errata) Bull. London Math. Soc. 15 (1983), no. 5, 401–487.
- MR 0648524. 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. ISBN 0-691-08304-5(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
- "The Geometry of 3-Manifolds (video)". Archived from the original on January 27, 2010. Retrieved January 20, 2010. A public lecture on the Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006.