Property R

Property R is a property of nodes which, according to Gabai's Property R theorem (formerly Property R conjecture ), applies to all nodes and which states that 0 surgery on a node only results in the unknot . This property is important in low-dimensional topology, for example when calculating Seiberg-Witten-Floer homology and Heegaard-Floer homology groups. ${\ displaystyle S ^ {2} \ times S ^ {1}}$

Unknot

motivation

Dehn surgery is a Returning to Max Dehn method for 3-dimensional manifolds structure by one selected from the 3-dimensional sphere node (or a multi-node entanglement ) drilled out and is otherwise glued again. According to Lickorish-Wallace's theorem , every closed 3-manifold is obtained by stretching surgery on a suitable loop.

In general, one and the same 3-manifold can be constructed in different ways by stretching surgery on different entanglements. For example, "trivial" surgery always returns the 3-sphere to any loop. ${\ displaystyle \ infty}$

On the other hand, the manifold can only be constructed in one way by means of 0 surgery on a knot, namely by means of 0 surgery on the unknot. This fact is known as property R , it was originally suggested by Valentin Poénaru in 1974 and proved by Gabai in 1987 . It implies some of Poénaru's conjectures about 3- and 4-manifolds. ${\ displaystyle S ^ {2} \ times S ^ {1}}$

Gabai's Property R theorem

Let be a knot and the 3-manifold obtained by stretching surgery on the knot with coefficient 0 (short: 0-surgery). ${\ displaystyle K \ subset S ^ {3}}$ ${\ displaystyle M_ {K} (0)}$ ${\ displaystyle K}$

If the unknot isn't, then it's not homeomorphic too . ${\ displaystyle K}$ ${\ displaystyle M_ {K} (0)}$ ${\ displaystyle S ^ {2} \ times S ^ {1}}$

Evidence Strategy

Gabai proves that there is a tight foliation of the nodal complement that cuts the marginal torus in longitudes . In the 3-manifold obtained by 0-surgery, these longitudes border disjoint circular disks, so one obtains a tight foliation of . With Novikov's theorem, it follows from this the irreducibility of , in particular is . ${\ displaystyle M_ {K} (0)}$ ${\ displaystyle M_ {K} (0)}$ ${\ displaystyle M_ {K} (0)}$ ${\ displaystyle M_ {K} (0) \ not = S ^ {2} \ times S ^ {1}}$

Other evidence comes from Gordon-Luecke and Parry.

generalization

Let be a knot and the 3-manifold obtained by stretching the knot with coefficients . ${\ displaystyle K \ subset S ^ {3}}$ ${\ displaystyle M_ {K} (r)}$ ${\ displaystyle K}$ ${\ displaystyle r \ in \ mathbb {Q}}$

If the unknot is not, then orientation-preserving is homeomorphic too . ${\ displaystyle K}$ ${\ displaystyle U}$ ${\ displaystyle M_ {K} (r)}$ ${\ displaystyle M_ {U} (r)}$

literature

Chapter 7.4 in Jennifer Schultens : Introduction to 3-manifolds. Graduate Studies in Mathematics, 151. American Mathematical Society, Providence, RI, 2014. ISBN 978-1-4704-1020-9

Individual evidence

^ David Gabai: Foliations and the topology of 3-manifolds. III. J. Differential Geom. 26 (1987) no. 3, 479-536.
↑ Cameron Gordon , John Edwin Luecke : Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989) no. 2, 371-415.
^ Walter Parry : All types implies torsion. Proc. Amer. Math. Soc. 110 (1990) no. 4, 871-875.
^ Peter Kronheimer , Tomasz Mrowka , Peter Ozsváth , Zoltán Szabó : Monopoles and lens space surgeries. Ann. of Math. (2) 165 (2007), no. 2, 457-546.

[1] David Gabai: Foliations and the topology of 3-manifolds. III. J. Differential Geom. 26 (1987) no. 3, 479-536.

[2] Cameron Gordon , John Edwin Luecke : Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989) no. 2, 371-415.

[3] Walter Parry : All types implies torsion. Proc. Amer. Math. Soc. 110 (1990) no. 4, 871-875.

[4] Peter Kronheimer , Tomasz Mrowka , Peter Ozsváth , Zoltán Szabó : Monopoles and lens space surgeries. Ann. of Math. (2) 165 (2007), no. 2, 457-546.