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


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


mathematics; millennium problems; Poincaré conjecture; Henri Poincaré


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 https://arxiv.org/abs/math/0211159

Perelman, G. (2003a). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. ArXiv. Retrieved from https://arxiv.org/abs/math/0307245

Perelman, G. (2003b). Ricci flow with surgery on three-manifolds. ArXiv. Retrieved from https://arxiv.org/abs/math/0303109

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.

Creative Commons License
Texts in the journal are –unless otherwise indicated– published under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License