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The geometry of 3-manifolds / Poincaré conjecture

n1vux on 2006-10-13T02:55:31

Earlier this week, I attended a public lecture in the Harvard "Research Lectures for Non-Specialsts" by Curtis McMullen (Harvard University): The geometry of 3-manifolds.

As I expected, this was a (relatively) elementary explication of the recent Poincaré conjecture progress, for which Grigori Pereleman was awarded one of the 2006 Fields Medals, and is in line for a share of the Clay Millennium Prize as well. As it happens, McMullen had previously won the Fields Medal himself, in part for cross-fertilizing techniques of chaotic-dynamics to the understanding of the larger Thurston's geometrization conjecture -- that Perelman solved (and declined the Medal). (Thurston had previously been awarded a Fields also.)

Thurston's conjecture seeks to categorize all finite, orientable, unbounded (hyper)surfaces (“manifolds”) according to their intrinsic geometries. Orientable leaves out the un-orientable Möbius strips and Klein bottles and their higher-dimensional analogues. Unbounded leaves out sheets of paper, paper-towel tubes, and again the Möbius strip. Finite leaves out an infinite plane, line, or space, or even a infinite spiral cylinder or library.

McMullen had both an inflated beach-ball and an inflated beach “inner tube”, both with marked equators, to demonstrate the topological difference of a sphere and a torus considered as 2-D manifolds. The Sphere passes the loop-test – any loop drawn on it's surface, including but not limited to all diameters, can be shrunk to a point and thus removed topologically from the sphere, but neither kind of diameter on a torus (slicing of bundt cake and of bagel) can be shrunk to a point and removed, only non-diameter loops can. This is an important distinction – of all the possible 2-dimensional surfaces (2-manifolds), the topologists' 2-sphere is the only one to pass the Loop test.

  • [Note on terminology - The topologists' 1-sphere (circle), 2-sphere (beachball surface), 3-sphere (hypersphere) are called by geometers the 2-sphere (closed line), 3-sphere (surface of a 3-ball), 4-sphere (hypersurface of a 4-ball). The Geometer's terminology was of course used in your high school geometry class. Why a difference? Geometers measure the embedding metric space. Topologists usually ignore embedding and look at the internal invariant metrics of the surface, such as it's own intrinsic dimension. Geometers would say hyper-surface explicitly if they meant a 3D hypersurface of a 4D-hyper-volume. *]


One of the key results of 2-D topology is there are only 3 potential geometries for a 2-D manifold. His illustration is of the three is [here]. – Spherical Geometry of course works on a sphere, amusingly the Euclidean geometry works on the torus, and the Hyperbolic geometry works for all higher genus 2-D manifolds (surfaces with more than one “handle”, places where a loop can be caught). Why does Hyperbolic work for all higher genus surfaces? All the extra room at the edges – see the huge number of angels and devils in the outer rings of the infinite regress in Escher's version, visible on his homepage (The text their on his homepage includes a quick overview of his topics.) Right-angle pentagons are a key to the hyperbolic plane.

Jumping up to 3-D hyper-surfaces – again finite but boundless – he used Dante's Paradisio/Inferno as his model of the 3-sphere, a nest of spheres proceeding from a point to a large sphere, and down to a small sphere again. (This 3-manifold is embeddable only in 4-space, not 3-space, obviously. Another common model is two 3-balls, meeting at all points on their exterior such that orientability is maintained. This is the hyper-surface of a 4-ball hyper-volume in 4-space.)

He then used the stage doors of the lecture hall to model a 3-D hyper-torus "hyper-surface" as near cube with sides identified and demonstrate that it too would be loop-linked as was the torus. If we went out the back exits, we'd reappear behind the projection screen; if we exited stage left, we'd reenter stage right. Digging through the floor, we'd drop in the ceiling. He pulled a cord from the door stage right across the stage to the stage left door, saying, if this were a 3-torus, I could reach out this door, back in the other door, to tie this end of the cord to the other end (which he seemingly does), then pull the loop through the space like this ... and the cord did pull through the door, and slack behind him disappeared out the stage-left door. He half convinced us we were indeed in a hyper-torus mini-universe and stuck there :-) , but he did demonstrate that there is no getting shrinking a loop on a 3-torus.

Original form of the Poincaré Conjecture is basically that the 3-sphere is the only closed finite 3-manifold that passes the loop test. The special case of the geometrization conjecture for 3-D requires 8 geometries, and requires allowing segmenting a manifold piece-wise into those 8 architectures. This conjecture includes Poincaré's implicitly, since the spherical geometry will pass the loop test, and the others won't – so if there are no other manifolds, only the unique spherical manifold would pass the loop. The other 7 geometries do not admit a pseudo-sphere, and the spherical geometry has only one possible manifold. Again, the hyperbolic geometry's extra “space” provides the flexibility to rigidly geometricize the stranger manifolds. The lovely "Hyperbolic Space tiled with Dodecahedra" is locally Euclidean – all those 90-90-90 rectilinear girder-and-beam intersections – but globally it has negative curvature. This is the 3-D analog to the hyperbolic tiling with right-angle pentagons, of course.

One of the many great mathematicians to have thought they had proved the Poincaré Conjecture said using Topological arguments was the way not to prove this topological conjecture; in retrospect, his self-criticism appears correct.

[I'm omitting the connection between Knot Theory and geometrization of their complement spaces, which is important in the history of the conjectures.]

Hamilton made great progress on the Poincaré-Thurston problems using “Ricci Flows”, to morph a manifold into it's smooth equivalent, but was stymied by singularities in dimensions higher than 2. McMullen showed a movie of Ricci Flow in action [probably this one. Ricci Flow deforms a curve or surface locally to reduce local curvature – and this has an emergent global property to create symmetry, and in 1-D curves embedded in 2-D space, as in the movie, this will never cause self-intersection.

But with 3-manifolds embedded in 4-space, it can cause the singularities that stymied Hamilton and others, and if repeated, that could fragment the manifold into droplets, which didn't seem useful. Perelman's key insight, according to McMullen , is that these singularities are features not bugs – they're the “cut here” points of the necessary partitioning of a peculiar manifold into it's different constituent metric geometries. Perelman added an analogue of physic's “entropy” to demonstrate that patching the singularities and continuing the Ricci Flow on the resulting pieces was not an infinite shattering but eventually converged to obvious geometries.

[Omitting discussion of singularlary democratic on-line referreing of posted pre-prints without actual paper publication.]

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References --

In Mathematics, Wolfram's Mathworld is more authoritative than WikiPedia, which may be more readable by non-mathematicians. http://mathworld.wolfram.com/Hypersphere.html >
http://en.wikipedia.org/wiki/Ricci_flow>
http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture >

Links for the Grigori Perelman Preprints at arXiv.org:

Image galleries used for his slides

  • His
  • Another


AMS summary of state of the proof -

"Notes on Perelman's Papers", Bruce Kleiner, John Lott,

"Ricci Flow and the Poincare Conjecture" John W. Morgan, Gang Tian, 473 pages with over 30 figures http://arxiv.org/abs/math/0607607>

Yau's slides http://www.mcm.ac.cn/Active/yau_new.pdf>

Humor - Perelman T-shirt