The Poincaré conjeture: a problem solved after a century of new ideas and continued

• Autores: María Teresa Lozano Imízcoz
• Localización: Mètode Science Studies Journal: Annual Review, ISSN 2174-3487, ISSN-e 2174-9221, Nº. 8, 2018 (Ejemplar dedicado a: Making Science. A multitude of perspectives), págs. 58-67
• Idioma: inglés
• DOI: 10.7203/metode.0.9265
• Enlaces
  • Texto completo
• Resumen

  • 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 pro-posed by Richard Hamilton.

• Referencias bibliográficas
  • 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...
  • 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,...
  • 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.