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

María Teresa Lozano Imízcoz


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.


topología; esfera; grupo fundamental; geometría riemanniana; flujo de Ricci

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Hamilton, R. (1982). Three-manifolds with positive Ricci curvature. Journal of Differential Geometry, 17(2), 255–306.

Jaco, W., & Shalen, P. B. (1978). A new decomposition theorem for irreducible sufficiently-large 3-manifolds. In J. Milgram (Ed.), Algebraic and geometric topology (pp. 71–84). Providence: American Mathematical Society. doi: 10.1090/pspum/032.2

Johannson, K. (1979). Homotopy equivalences of 3-manifolds with boundaries. Berlin: Springer-Verlag.

Kneser, H. (1929). Geschlossene Flächen in dreidimesnionalen Mannigfaltigkeiten. Jahresbericht der Deutschen Mathematiker-Vereinigung, 38, 248–260.

Milnor, J. (1962). A unique decomposition theorem for 3-manifolds. American Journal of Mathematics, 84(1), 1–7.

O’Shea, D. (2007). The Poincaré conjecture: In search of the shape of the universe. New York: Walker Publishing Company.

Perelman, G. (2002). The entropy formula for the Ricci flow and its geometric applications. ArXiv. Retrieved from

Perelman, G. (2003a). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. ArXiv. Retrieved from

Perelman, G. (2003b). Ricci flow with surgery on three-manifolds. ArXiv. Retrieved from

Poincaré, H. (1904). Cinquième complément à l’analysis situs. Rendiconti del Circolo Matematico di Palermo, 18(1), 45–110.

Scott, P. (1983). The geometries of 3-manifolds. Bulletin of the London Mathematical Society, 15(5), 401–487


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