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.¹ 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.² If a knot ${\displaystyle K\subset \mathbb {S} ^{3}}$ has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along ${\displaystyle K}$.

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

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

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