Property P conjecture

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 R. H. 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.

See also

• Property R conjecture

References

• Eliashberg, Yakov (2004). "A few remarks about symplectic filling". Geometry & Topology. 8: 277–293. arXiv:math.SG/0311459. doi:10.2140/gt.2004.8.277.
• Etnyre, John B. (2004). "On symplectic fillings". Algebraic & Geometric Topology. 4: 73–80. arXiv:math.SG/0312091. doi:10.2140/agt.2004.4.73.
• Kronheimer, Peter; Mrowka, Tomasz (2004). "Witten's conjecture and Property P". Geometry & Topology. 8: 295–310. arXiv:math.GT/0311489. doi:10.2140/gt.2004.8.295.
• Ozsvath, Peter; Szabó, Zoltán (2004). "Holomorphic disks and genus bounds". Geometry & Topology. 8: 311–334. arXiv:math.GT/0311496. doi:10.2140/gt.2004.8.311.