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

• Grigori Perelman, The entropy formula for the Ricci flow and its geometric applications (arXiv:math/0211159)

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

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)