Property P

Property P is a property of knots that, according to the property P conjecture proven by Peter Kronheimer and Tomasz Mrowka in 2004, applies to all nontrivial knots and that 1-surgery on the knot never results in a homotopy sphere. This conjecture was important in the development of the 3-dimensional topology, especially in connection with the Poincaré conjecture .

Property P

Let be a knot and the 3-manifold obtained by stretching surgery on the knot with coefficient 1 (short: 1-surgery) . ${\displaystyle K\subset S^{3}}$ ${\displaystyle M_{K}(1)}$ ${\displaystyle K}$

${\displaystyle K}$has property P if the 3-manifold created by 1-surgery has a nontrivial fundamental group . ${\displaystyle \pi _{1}(M_{K}(1))\not =0}$

Kronheimer-Mrowka's theorem says: If the unknot is not, then it has the property P. ${\displaystyle K}$

history

Property P has been discussed in connection with work on the Poincaré conjecture since the 1950s, around 1958 in a paper by Bing.

The "Property P Conjecture" in its more general version stated that nontrivial stretching surgery on a nontrivial knot can never produce a homotopy sphere. (A nontrivial stretching surgery means a surgery with coefficient .) This conjecture was made in the 1970s by RH Bing and Martin and independently of González-Acuña as a step towards proving the Poincaré conjecture. ${\displaystyle {\tfrac {p}{q}}\not ={\tfrac {1}{0}}}$

According to Gordon-Luecke's theorem , which was proven at the end of the 1980s , knots are clearly determined by their complement, in particular a nontrivial stretching surgery on a nontrivial knot can never give that. From the Poincaré conjecture it would follow that every nontrivial node has property P. ${\displaystyle S^{3}}$

With the Culler-Gordon-Luecke-Shalen theorem on cyclic surgery proven in 1987 , the general version (about any nontrivial stretching surgery) can be reduced to the more specific formulation given above (about 1-surgeries), i.e. the problem is reduced to the proof of to reduce. ${\displaystyle \pi _{1}(M_{K}(1))\not =0}$

For the proof of one tried to construct nontrivial representations of in suitable Lie groups , e.g. SU (2) . In this context, the development of the Casson invariant and the Instanton-Floer homology was important. ${\displaystyle \pi _{1}(M_{K}(1))\not =0}$ ${\displaystyle \pi _{1}(M_{K}(1))}$

Property P for arbitrary nodes (more precisely: the existence of a non-trivial homomorphism ) was proven by Kronheimer-Mrowka in 2004 with the help of Seiberg-Witten invariants as well as existence theorems for scrolls and contact structures . It also follows from Perelman's proof of the Poincaré conjecture. ${\displaystyle \pi _{1}(M_{K}(1))\to SO(3)}$

See also

Property R

literature

J. Rasmussen: Review to online

Individual evidence

1. ↑ RH Bing: Necessary and sufficient conditions that a 3-manifold be S ³ . Ann. of Math. (2) 68 1958 17-37.
2. ↑ M. Culler, C. Gordon, J. Luecke, P. Shalen: Dehn surgery on knots. Ann. of Math. (2) 125 (1987) no. 2, 237-300.
3. ↑ P. Kronheimer, T. Mrowka: Witten's conjecture and property P. Geom. Topol. 8: 295-310 (2004).