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The Poincaré conjecture is a major theorem in [[algebraic topology]] first proposed in 1904 by [[Henri Poincaré]] and was not proven until the solution was posted by [[Best of the public|Grigori Perelman]] on the [[internet]] beginning in 2002.

The Poincaré conjecture is a major theorem in [[algebraic topology]] first proposed in 1904 by [[Henri Poincaré]] and was not proven until the solution was posted by [[Best of the public|Grigori Perelman]] on the [[internet]] beginning in 2002.

==Layman Statement of the Conjecture==

==Layman statement of the conjecture==

An ordinary sphere (such as the surface of orange) is "[[fundamental group|simply connected]]" because a stretchable loop on it can be reduced to a single point without tearing it.  A doughnut is an example of an object that is not simply connected.

An ordinary sphere (such as the surface of orange) is "[[fundamental group|simply connected]]" because a stretchable loop on it can be reduced to a single point without tearing it.  A doughnut is an example of an object that is not simply connected.

Put another way, the conjecture states that if strings, surfaces, and other geometric objects "loop around" a three dimensional space in the same way they loop around what mathematicians call a "three sphere" (see below), then the space in question is a three sphere itself.

Put another way, the conjecture states that if strings, surfaces, and other geometric objects "loop around" a three dimensional space in the same way they loop around what mathematicians call a "three sphere" (see below), then the space in question is a three sphere itself.

===The Three Sphere===

===The three sphere===

Mathematicians call the ordinary circle the "one sphere," which is written <math>\mathbb{S}^1 \ </math>.  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 <math>\mathbb{S}^1 \ </math> - 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 <math>\mathbb{S}^1 \ </math>.

Mathematicians call the ordinary circle the "one sphere," which is written <math>\mathbb{S}^1 \ </math>.  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 <math>\mathbb{S}^1 \ </math> - 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 <math>\mathbb{S}^1 \ </math>.

The three sphere, or <math>\mathbb{S}^3 \ </math>, 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 <math>\mathbb{S}^3 \ </math>.

The three sphere, or <math>\mathbb{S}^3 \ </math>, 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 <math>\mathbb{S}^3 \ </math>.

===Strings, Surfaces, and Looping Around===

===Strings, surfaces, and looping around===

:''For more information please see:'' [[Fundamental group]] and [[Homotopy group]]

:''For more information please see:'' [[Fundamental group]] and [[Homotopy group]]

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 Poincaré conjecture, with the understanding that in addition to loops of string, we also discuss "tying" spheres around a space.

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 Poincaré conjecture, with the understanding that in addition to loops of string, we also discuss "tying" spheres around a space.

===Formal Statement of the Conjecture===

===Formal statement of the conjecture===

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

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

::''Let <math>X \ </math> be an <math>n \ </math>-dimensional manifold.  Suppose that for every <math>m \leq n \ </math>, <math>\pi_m(X) \cong \pi_m(\mathbb{S}^n) \ </math>.  Does it follow that <math>X \ </math> is homeomorphic to <math>\mathbb{S}^n \ </math>?

::''Let <math>X \ </math> be an <math>n \ </math>-dimensional manifold.  Suppose that for every <math>m \leq n \ </math>, <math>\pi_m(X) \cong \pi_m(\mathbb{S}^n) \ </math>.  Does it follow that <math>X \ </math> is homeomorphic to <math>\mathbb{S}^n \ </math>?

===Homeomorphism v. Diffeomorphism===

===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 <math>\leq 3 \ </math>, and it turns out that a diffeomorphism exists for <math>n=5 \ </math> and <math>n=6 \ </math>, but for most <math>n\geq 7 \ </math> it is possible to construct objects which are homeomorphic and not diffeomorphic - such objects are called "exotic <math>\mathbb{S}^n</math>." The only open case is the four dimensional one.

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 <math>\leq 3 \ </math>, and it turns out that a diffeomorphism exists for <math>n=5 \ </math> and <math>n=6 \ </math>, but for most <math>n\geq 7 \ </math> it is possible to construct objects which are homeomorphic and not diffeomorphic - such objects are called "exotic <math>\mathbb{S}^n</math>." The only open case is the four dimensional one.

==Proof of the Conjecture==

==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.

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===

===Easy cases===

While the original statement of the conjecture, also called the <math>n=3 \ </math> case, went unproved for a century, other versions were easily proved.

While the original statement of the conjecture, also called the <math>n=3 \ </math> case, went unproven for a century, other versions were easily proven.

<math>n=2 \ </math> 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 <math>n=2 \ </math> case.

<math>n=2 \ </math> was proven 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 <math>n=2 \ </math> case.

<math>n \geq 5 \ </math> was proved by Steve Smale in 1961 with his h-cobordism theorem, for which he won a [[Fields Medal]].   

<math>n \geq 5 \ </math> was proved by Steve Smale in 1961 with his h-cobordism theorem, for which he won a [[Fields Medal]].   

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 Poincaré's original formulation of the conjecture open.

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 Poincaré's original formulation of the conjecture open.

===Perelman's Proof===

===Perelman's proof===

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

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