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Solution of the Poincaré conjecture

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In November 2002, Grigori Perelman posted the first of a series of eprints on arXiv outlining a solution of the Poincaré conjecture. Perelman's proof uses a modified version of a Ricci flow program developed by Richard Hamilton. On March 18, 2010, the Clay Mathematics Institute awarded Perelman the Millennium Prize in recognition of his proof.[1]

Millennium Prize Problems
P versus NP problem
Hodge conjecture
Poincaré conjecture (solution)
Riemann hypothesis
Yang–Mills existence and mass gap
Navier–Stokes existence and smoothness
Birch and Swinnerton-Dyer conjecture

Contents

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Description

The Poincaré conjecture says that if a 3-dimensional manifold is compact, has no boundary and is simply connected, then it is homeomorphic to a 3-dimensional sphere. The concepts of "manifold", "compact", "no boundary", "simply connected", "homeomorphic" and "3-dimensional sphere" are described below. Perelman (using ideas originally from Hamilton) proved the conjecture by deforming the manifold using something called the Ricci flow (which behaves similarly to the heat equation that describes the diffusion of heat through an object). The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself (like hot mozzarella) towards what are known as singularities. Perelman and Hamilton then chop the manifold at the singularities (a process called "surgery") causing the separate pieces to form into ball-like shapes. Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and wondering whether the surgery might need to be repeated infinitely many times.

Explaining the key terms

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What is a 3-dimensional sphere?

A one-dimensional sphere is a circle, which can be thought of as the set of points, (x, y), in two dimensions that satisfy the equation x2 + y2 = r2, where r is the radius. A two-dimensional sphere is the surface of a globe, or the set of points, (x, y, z) in three dimensions that satisfy the equation x2 + y2 + z2 = r2. And a three-dimensional sphere is the set of points in four dimensions, (x, y, z, w), that satisfy the equation x2 + y2 + z2 + w2 = r2.

Circle is a one dimensional sphere

The boundary of a common ball is the 2-dimensional sphere.

The 3-sphere is one dimension higher, the boundary of a four-dimensional ball

What is a manifold?

A manifold is a surface created by taking another surface -- for example, a piece of paper -- and warping it. A cylinder is a manifold since it can be formed by attaching the two opposite sides of the paper to each other. The cylinder can be deformed into another manifold by attaching the two circles at each end of the cylinder, to get a torus (ie. donut).

A manifold is a space created by gluing together pieces of Euclidean space, called charts. For example you could take two 2-dimensional disks and curve them around two hemispheres and then glue them together to make a 2-dimensional sphere.

Hemispheres mapped together to form a Sphere

A torus (the surface of a donut) could be built using a rectangular chart as seen in this image.

Progressive build of a torus from a rectangle

A 3-dimensional sphere can be made using a pair of solid 3-dimensional balls: identify each point of the boundary of the first ball with the corresponding point of the second ball.

Other manifolds can be created in similar ways. See manifold for an easy and advanced description. Manifolds can be warped or distorted using diffeomorphisms.

What does no boundary mean?

We say a manifold has an edge or a boundary if one of the charts is not glued to another on all sides. One of the conditions in the Poincaré conjecture is that there be no edges, just like in the sphere and the torus.

What does compact mean?

A compact manifold is bounded and does not extend to infinity. Both an infinitely long cylinder and an infinite plane are examples of manifolds which are not compact. In Poincaré's conjecture it is required that the manifolds be compact. See compact space for an advanced definition.

What does simply connected mean?

A manifold is simply connected if any loop drawn on the space can be deformed to a point without leaving the manifold. Any line drawn on a simply connected manifold that starts and ends at the same point can be shrunk down to one point without any part of it leaving the manifold. A torus is not simply connected, since you can draw a loop going around the cylinder that you can't contract to a point without taking it off.

An example of a simply connected manifold is a sphere: if you try to wrap a lasso around a sphere it will slide off. An example of a manifold which is not simply connected is a torus. One can tie a lasso around a donut and catch hold of it. Nothing short of untying the lasso or cutting the donut will let it loose. See simply connected for an easy and advanced description.

A loop on a sphere can be contracted to a point without leaving the surface.
Neither of the colored loops on this torus can be contracted to a point without leaving the surface.

What does homeomorphic mean?

Generally, two shapes are homeomorphic if you can deform one into the other without a break or discontinuity. A homeomorphism is a continuous function mapping points from one object to another. This means that if two points are close to each other in the first object, they will be close together when the points are mapped onto the second object. The surface of a square and the surface of a sphere are not homeomorphic, since the square has edges and the sphere doesn't, so the mapping function has to jump somewhere, and at that point it won't be continuous.

A homeomorphism between two spaces, A and B, is a correspondence between the points of A and B, such that each point of A corresponds to exactly one point of B and vice versa, which is a continuous function both when viewed from A to B and from B to A. Intuitively, this means that if two points are close to each other in A, the corresponding points in B are also close to each other, and vice versa. Two spaces are called homeomorphic if a homeomorphism between them exists.

For example, a 2-dimensional sphere is homeomorphic to the surface of a cube; similarly, a 3-dimensional sphere is homeomorphic to the 3-dimensional boundary of a 4-dimensional hypercube.

Putting all these terms together, we can now understand the statement of the Poincaré conjecture:

The Poincaré conjecture says that a 3-dimensional manifold which is compact, has no boundary and is simply connected must be homeomorphic to a 3-dimensional sphere.

Perelman's proof based on Hamilton's Ricci flow

The first step is to deform the manifold using the Ricci flow. The Ricci flow was used by Richard Hamilton as a way to deform manifolds. He used it to prove that many compact manifolds were diffeomorphic to spheres. However, he did not prove they were all diffeomorphic to spheres. The Ricci flow is an imitation of the heat equation which describes the way heat flows in a solid. Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow, the Ricci flow could run into singularities and stop functioning.

Hamilton was able to list a number of possible singularities that could form but he was concerned as to whether he had found all possible singularities. He wanted to cut the manifold at the singularities and paste in caps, and then run the Ricci flow again. But he needed to understand the singularities. Grigori Perelman examined the singularities and discovered they were very simple manifolds: essentially three-dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking circles stretched along a line.

This was proved using something Perelman called the "Reduced Volume" which is closely related to an eigenvalue of a certain "elliptic equation". Eigenvalues are difficult to describe without calculus but they are part of a famous problem: Can you hear the shape of a drum?. Essentially an eigenvalue is like a note being played by the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helped him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the cigar solution, which looked like a strand sticking out of a manifold with nothing on the other side. In essence Perelman showed that all the strands that form can be cut and capped and none stick out on one side only.

Completing the proof, Perelman takes any compact, simply connected, three-dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them. He cuts the strands and continues deforming the manifold until eventually he is left with a collection of round three-dimensional spheres. Then he rebuilds the original manifold by connecting the spheres together with three-dimensional cylinders, morphs them into a round shape and sees that, despite all the initial confusion, the manifold was in fact diffeomorphic to a sphere.

Two immediate questions were then: how can one be sure there aren't infinitely many cuts necessary? That the cutting does not progress forever? Perelman proved this using soap films on the manifold and showing that the areas of the soap films decreases as the manifold undergoes Ricci flow. Eventually the area is so small that any cut after the area is that small can only be chopping off three-dimensional spheres and not more complicated pieces. This is described as a battle with a Hydra in Szpiro's book cited below.

References

  1. ^ Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman

Research papers

See History of Perelman's proof and elaborations for detailed chronology.

Perelman's papers

Perelman's original papers containing the proof:

  • Perelman, Grisha (November 11, 2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. 
  • Perelman, Grisha (March 10, 2003). Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109. 
  • Perelman, Grisha (July 17, 2003). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245. 

Detailed

  • Bruce Kleiner, John Lott Notes on Perelman's papers arXiv:math/0605667
  • Huai-Dong Cao, Xi-Ping Zhu (December 3, 2006). Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture. arXiv:math.DG/0612069. 
  • John W. Morgan, Gang Tian Ricci Flow and the Poincaré Conjecture arXiv:math/0607607
Detailed proof, expanding Perelman's papers.

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