geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...

, the Property P conjecture is a statement about

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

s obtained by

Dehn surgery In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...

on a

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

in the

3-sphere In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...

. 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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoint ...

. 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 In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured b ...

, since the Lickorish–Wallace theorem says any closed, orientable 3-manifold results from Dehn surgery on a link. If a knot

K \subset \mathbb^

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

\in \pi_(\mathbb^ \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^

for some

0 \ne a \in \mathbb

References

* * * * * * 3-manifolds Conjectures that have been proved {{topology-stub