Property P conjecture

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 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.

ee also

*Property R conjecture

References

*Yakov Eliashberg, [http://dx.doi.org/10.2140/gt.2004.8.277 "A few remarks about symplectic filling"] , Geometry and Topology 8 (2004) 277-293 arXiv: [http://www.arxiv.org/abs/math.SG/0311459 math.SG/0311459]
*John B. Etnyre, [http://dx.doi.org/10.2140/agt.2004.4.73 "On symplectic fillings"] , Algebraic and Geometric Topology 4 (2004) 73-80 arXiv: [http://www.arxiv.org/abs/math.SG/0312091 math.SG/0312091]
*Peter Kronheimer, Tomasz Mrowka, [http://dx.doi.org/10.2140/gt.2004.8.295 "Witten's conjecture and Property P"] , Geometry and Topology 8 (2004) 295-310 arXiv: [http://www.arxiv.org/abs/math.GT/0311489 math.GT/0311489]
*Peter Ozsvath, Zoltan Szabo, [http://dx.doi.org/10.2140/gt.2004.8.311 "Holomorphic disks and genus bounds"] , Geometry and Topology 8 (2004) 311-334 arXiv: [http://www.arxiv.org/abs/math.GT/0311496 math.GT/0311496]


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