nLab Poincaré conjecture

Statement

Theorem

(Poincaré conjecture)

Every simply connected compact topological 3-manifold without boundary is homeomorphic to the 3-sphere.

Proof strategy

A proof strategy was given by Richard Hamilton: imagine the manifold is equipped with a metric. Follow the Ricci flow of that metric through the space of metrics. As the flow proceeds along parameter time, it will from time to time pass through metrics that describe singular geometries where the compact metric manifold pinches off into separate manifolds. Follow the flow through these singularities and then continue the flow on each of the resulting components. If this process terminates in finite parameter time with the metric on each component stabilizing to that of the round 3-sphere, then the original manifold was a 3-sphere.

The hard technical part of this program is to show that the passage through the singularities can be controlled. This was finally shown in (Perelman 02).

See at Ricci flow for more.

Related entries

• geometrization conjecture

References

In dimension 3

The proof of the Poincaré conjecture is attributed to these unpublished articles:

• Grigori Perelman: The entropy formula for the Ricci flow and its geometric applications [arXiv:math/0211159]

• Grigori Perelman: Ricci flow with surgery on three-manifolds [arXiv:0303109]

• Grigori Perelman: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds [arXiv:math/0307245]

Further discussion:

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

• John W. Morgan, Gang Tian: Ricci Flow and the Poincaré Conjecture, Clay Mathematics Monographs 3 (2007) [arXiv:math/0607607, webpage, pdf]

Notes from a survey talk:

• Huisken on Uniformization I

• Huisken on Uniformization II

See also

• Bruno Martelli, An Introduction to Geometric Topology (arXiv:1610.02592)

In higher dimesnions

On the analog of the Poincaré conjecture in higher dimensions:

• M. H. A. Newman, Theorem 7 in: The Engulfing Theorem for Topological Manifolds, Annals of Mathematics Second Series, 84* 3 (1966) 555-571 (jstor:1970460)

• Laurent Siebenmann: La conjecture de Poincaré topologique en dimension 4, Séminaire Bourbaki volume 1981/82, exposés 579-596, Astérisque, no. 92-93 (1982), Talk no. 588 [numdam:SB_1981-1982__24__219_0]

• Laurent Siebenmann (translation by M. H. Kim & M. Powell): Topological Poincaré conjecture in dimension 4 (the work of M. H. Freedman), Celebratio Mathematica [celebratio:752]