Poincaré conjecture: A problem solved after a century of new ideas and continued work

Abstract

The Poincare conjecture is a topological problem established in 1904 by the French mathematician Henri Poincare. It characterises three-dimensional spheres in a very simple way. It uses only the first invariant of algebraic topology – the fundamental group – which was also defined and studied by Poincare. The conjecture implies that if a space does not have essential holes, then it is a sphere. This problem was directly solved between 2002 and 2003 by Grigori Perelman, and as a consequence of his demonstration of the Thurston geometrisation conjecture, which culminated in the path proposed by Richard Hamilton.