nLab geometrization conjecture

Idea

Every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure (…)

The proof of the geometrization conjecture implies that of

• the Poincaré conjecture

• the elliptization conjecture

Related entries

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References

The conjecture was stated by William Thurston in 1982.

Review:

• Peter Scott, section 6 of: The Geometries of 3-Manifolds, Bulletin of the LMS 15 5 (1983) 401-487 [doi:10.1112/blms/15.5.401, hdl:2027.42/135276]

The Ricci flow-strategy for the proof was devised by Richard Hamilton and the remaining technical problem of controlling the pinching was finally solved by Grigori Perelman.

See also:

• Wikipedia, Geometrization conjecture

• Laurent Bessieres, Gerard Besson, Michel Boileau, Sylvain Maillot, Joan Porti, Geometrisation of 3-manifolds (pdf)

Discussion for 3-dimensional orbifolds:

• Michel Boileau, Bernhard Leeb, Joan Porti, Geometrization of 3-Dimensional Orbifolds, Annals of Mathematics Second Series, Vol. 162, No. 1 (2005), pp. 195-290 (jstor:3597373)

• Bruce Kleiner, John Lott, Geometrization of Three-Dimensional Orbifolds via Ricci Flow (arXiv:1101.3733)