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 not simply-connected. The conjecture states that all knots, except the unknot, have Property P.

Research on Property P was 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. If a knot has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along .

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

Algebraic Formulation

Let denote elements corresponding to a preferred longitude and meridian of a tubular neighborhood of .

has Property P if and only if its Knot group is never trivialised by adjoining a relation of the form for some .

gollark: ```This egg feels like all future spacetime trajectories lead into it.```
gollark: GϘn.
gollark: Revised description:```Mana courses through this glassy egg, producing a beautiful glow - it's very reflective, almost metallic. It has a red gleam, too, and smells faintly like brine. It shimmers like gold, and it seems as if time is distorted around it. It is much smaller than the other eggs, and looks like lots of pieces of paper folded together and smelling faintly like cheese. It occupies every point in the spacetime continuum.```
gollark: Oh, forgot it.
gollark: Reminder: they'll all be omnidragons.

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.
  • Rolfsen, Dale (1976), "Chapter 9.J", Knots and Links, Mathematics Lecture Series, 7, Berkeley, California: Publish or Perish, pp. 280–283, ISBN 0-914098-16-0, MR 0515288
  • Adams, Collin. The Knot Book : An elementary introduction to the mathematical theory of knots. American Mathematical Society. p. 262. ISBN 0-8218-3678-1.
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