Property P conjecture
Template:Short description Template:Inline-citationsIn 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.<ref>Template:Cite thesis</ref> If a knot <math>K \subset \mathbb{S}^{3}</math> has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along <math>K</math>.
A proof was announced in 2004, as the combined result of efforts of mathematicians working in several different fields.
Algebraic FormulationEdit
Let <math>[l], [m] \in \pi_{1}(\mathbb{S}^{3} \setminus K)</math> denote elements corresponding to a preferred longitude and meridian of a tubular neighborhood of <math>K</math>.
<math>K</math> has Property P if and only if its Knot group is never trivialised by adjoining a relation of the form <math> m = l^{a} </math> for some <math> 0 \ne a \in \mathbb{Z}</math>.