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


Abstract


The Poincaré conjecture is a topological problem established in 1904 by the French mathematician Henri Poincaré. 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 Poincaré. 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.

Keywords


topology; sphere; fundamental group; Riemannian geometry; Ricci flow

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References


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