Poincaré conjecture

The Poincare conjecture is a major theorem in algebraic topology first proposed in 1904 by Henri Poincaré and not proved until 2006 by Gregori Perelman.

Layman Statement of the Conjecture

The basic idea of the conjecture is that if strings, surfaces, and other geometric objects "loop around" a three dimensional space space in the same way they loop around what mathematicians call a "three sphere," then the space in question is a three sphere itself.

The Three Sphere

Mathematicians call the ordinary circle the "one sphere," which is written Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}^1 \ } . The defining quality of a circle is that it is the set of all points in plane which are the same distant from a specific point. Any deformation of this circle is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}^1 \ } - an object does not need to be a perfect circle to be called a one sphere by topologists, any closed loop which does not intersect itself is an example of $\mathbb {S} ^{1}\$.

The surface of an ordinary sphere, like the Earth, is called a two sphere, or $\mathbb {S} ^{2}\$. The defining quality of a sphere is that it is the set of all points in a Euclidean (ie, normal) three dimensional space which are the same distance from a set point. Note that the interior of a sphere is not $\mathbb {S} ^{2}\$, and that like $\mathbb {S} ^{1}\$, a deflated beach ball stretched into some strange shape is still $\mathbb {S} ^{2}\$, so long as it does not self-intersect.

The three sphere, or $\mathbb {S} ^{3}\$, refers to the set of all points in a four dimensional Euclidean space which are the same distance from a single point. Any deformation of that object is a $\mathbb {S} ^{3}\$.

Strings, Surfaces, and Looping Around

For more information please see: Homotopy group

By strings and surfaces, we mean the possible ways of embedding $\mathbb {S} ^{1}\$ and $\mathbb {S} ^{2}\$ into other spaces.

For the embedding of strings ($\mathbb {S} ^{1}\$) into two-dimensional spaces, it is easiest to imagine holding strings on these surfaces. For example, it is clear that any string we tie around a sphere, a basketball, for example, can be contracted until it is very small, even a point. While we can drape a string on a doughnut so that it can be shrunk to a point, there are ways we can loop strings around a doughnut so that they cannot - through the hole in the center, for example. Thus, we can distinguish between a sphere and a doughnut by noting that we can tie strings around them in different ways.

The natural question arises, if we CANNOT tie strings around two objects in different ways, are they the same object? This is, in essence, the Poincare conjecture, with the understanding that in addition to loops of string, we also discuss "tying" spheres around a space.

Formal Statement of the Conjecture

For those readers who have taken undergraduate topology, the statement of the Poincare conjecture is much shorter:

Let $X\$ be a $3\$-dimensional manifold. Suppose that for every $m\leq 3\$, $\pi _{m}(X)\cong \pi _{m}(\mathbb {S} ^{3})\$. Does it follow that $X\$ is homeomorphic to $\mathbb {S} ^{3}\$?

There is a more general form, which replaces $3\$ with $n\$:

Let $X\$ be an $n\$-dimensional manifold. Suppose that for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \leq n \ } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_m(X) \cong \pi_m(\mathbb{S}^n) \ } . Does it follow that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \ } is homeomorphic to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}^n \ } ?

Homeomorphism v. Diffeomorphism

For those readers who have some experience with topology, there may be a question of whether a diffeomorphism always exists. While the existence of a homeomorphism between manifolds demonstrates the existence of a diffeomorphism between them if they are of dimension Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq 3 \ } , the h-cobordism theorem actually demonstrates that a diffeomorphism exists for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\geq 5 \ } . The only open case is the four dimensional one.

Proof of the Conjecture

The original statement of the conjecture went unproved until Gregori Perelman, a relatively unknown Russian mathematician, completed the proof which had eluded a century of expert mathematicians by demonstrating that no "cigar singularities" form in the Ricci flow on manifolds satisfying the conditions of the conjecture.

Easy Cases

While the original statement of the conjecture, also called the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=3 \ } case, went unproved for a century, other versions were easily proved.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=2 \ } was proved even before the problem was stated by the easy classification of two-manifolds. Any senior-year college student who is majoring in mathematics should be able to prove the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=2 \ } case.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq 5 \ } was proved by Steve Smale in 1961 with his h-cobordism theorem, for which he won a Fields Medal.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=4 \ } was proved in 1981 by M. Freedman with his theorem on Casson handles, for which he shared a Fields medal with A Casson, who had pioneered study in this area. Freedman's theorem was essentially a strengthening of the results of Smale's cobordism theorem, obtained at the expense of weakening the premises.

Unfortunately, the so-called "Whitney trick" which allowed the cobordism theorems of the late 20th century to produce such breakthroughs in topology is not possible in three dimensions, leaving Poincare's original formulation of the conjecture open.

Perelman's Proof

Perelman's proof of the conjecture relied on a concept called "Ricci flow," which is time-dependent deformation of a space which expands areas which are curved "negatively" (ie, two particles traveling in the same direction diverge) and contracts area which are curved "positively" (ie, two particles traveling in the same direction converge).

The effect of the Ricci flow has much in common, mathematically at least, with the diffusion of heat in a surface, a concept called entropy.