In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions A Dehn surgery is a specific construction used to modify 3-manifolds The process takes as input a 3-manifold together with a link. In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations In Mathematics, a 3-sphere is a higher-dimensional analogue of a 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 non-simply-connected. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be The conjecture states that all knots, except the unknot, have Property P.
Research on Property P was jump-started by RH Bing, who popularized the name and conjecture. RH Bing (October 20 1914 - April 28 1986 was an influential American Mathematician.
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. In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among In Mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any Closed, Orientable, connected 3-manifold may be
A proof was announced in 2004, as the combined result of efforts of mathematicians working in several different fields.
See also
- Property R conjecture
References
- Yakov Eliashberg, A few remarks about symplectic filling, Geometry and Topology 8 (2004) 277-293 arXiv:math.SG/0311459
- John B. Geometry & Topology (ISSN 1364-0380 online 1465-3060 printed is a peer-refereed international Mathematics research journal devoted to Geometry and The arXiv ( pronounced " Archive " as if the "X" were the Greek letter Chi, χ is an Archive for electronic Etnyre, On symplectic fillings, Algebraic and Geometric Topology 4 (2004) 73-80 arXiv:math.SG/0312091
- Peter Kronheimer, Tomasz Mrowka, Witten's conjecture and Property P, Geometry and Topology 8 (2004) 295-310 arXiv:math.GT/0311489
- Peter Ozsvath, Zoltan Szabo, Holomorphic disks and genus bounds, Geometry and Topology 8 (2004) 311-334 arXiv:math.GT/0311496
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