Property P conjecture

In geometric topology, 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 $K\subset \mathbb {S} ^{3}$ has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along $K$.

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

Algebraic Formulation

Let $[l],[m]\in \pi _{1}(\mathbb {S} ^{3}\setminus K)$ denote elements corresponding to a preferred longitude and meridian of a tubular neighborhood of $K$.

$K$ has Property P if and only if its Knot group is never trivialised by adjoining a relation of the form $m=l^{a}$ for some $0\neq a\in \mathbb {Z}$.

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, vol. 7, Berkeley, California: Publish or Perish, pp. 280–283, ISBN 0-914098-16-0, MR 0515288
• Adams, Colin (2004). The Knot Book : An elementary introduction to the mathematical theory of knots. American Mathematical Society. p. 262. ISBN 0-8218-3678-1.