Poincaré conjecturemathematics

Assorted References

• major reference ( in topology: Fundamental group )

  The most famous of these questions, called the Poincaré conjecture, asks if a compact three-dimensional manifold with trivial fundamental group is necessarily homeomorphic to the three-dimensional sphere (the set of points in four-dimensional space that are equidistant from the origin), as is known to be true for the two-dimensional case. Much research in algebraic topology has been...

• development by Poincaré ( in Poincaré, Henri )

  ...cannot. Poincaré asked if a three-dimensional manifold in which every curve can be shrunk to a point is topologically equivalent to a three-dimensional sphere. This problem (now known as the Poincaré conjecture) became one of the most important unsolved problems in algebraic topology. Ironically, the conjecture was first proved for dimensions greater than three: in dimensions five...

• history of mathematics ( in mathematics: Algebraic topology )

  On this base, conjectures were made and a general theory produced, first by Poincaré and then by the American engineer-turned-mathematician Solomon Lefschetz, concerning the nature of manifolds of arbitrary dimension. Roughly speaking, a manifold is the n-dimensional generalization of the idea of a surface; it is a space any small piece of which...

• solution in four dimensions ( in Freedman, Michael Hartley )

  American mathematician who was awarded the Fields Medal in 1986 for his solution of the Poincaré conjecture in four dimensions.

• work of Thurston ( in Thurston, William Paul )

  ...manifold is locally isometric to just one of a family of eight distinct types. Special cases were then proved, but only in 2006 was the first generally convincing proof published of the Poincaré conjecture in three dimensions, which was a major unresolved part of Thurston’s geometrization conjecture. Grigori Perelman was awarded a Fields Medal in 2006 for this achievement,...