Property P conjecture

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In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-trivial) Dehn surgery on the knot is non-simply-connected. The conjecture states that all knots, except the unknot, have Property P.

Research on Property P was jump-started by RH Bing, who popularized the name and conjecture.

This conjecture can be thought of as a first step to resolving the Poincaré conjecture, since the Lickorish-Wallace theorem says any closed, orientable 3-manifold results from Dehn surgery on a link.

A proof was announced in 2004, as the combined result of efforts of mathematicians working in several different fields.

[edit] See also

• Property R conjecture

[edit] References

• Yakov Eliashberg, A few remarks about symplectic filling, Geometry and Topology 8 (2004) 277-293 arXiv:math.SG/0311459
• John B Etnyre, On symplectic fillings, Algebraic and Geometric Topology 4 (2004) 73-80 arXiv:math.SG/0312091
• Peter Kronheimer, Tomasz Mrowka, Witten's conjecture and Property P, Geometry and Topology 8 (2004) 295-310 arXiv:math.GT/0311489
• Peter Ozsvath, Zoltan Szabo, Holomorphic disks and genus bounds, Geometry and Topology 8 (2004) 311-334 arXiv:math.GT/0311496