Stretching surgery

In topology , a branch of mathematics , Dehn surgery is a method for the construction of 3-dimensional manifolds , which goes back to Max Dehn , by drilling a node out of the 3-dimensional sphere and gluing it back in another way.

A knot in 3-dimensional space.

A full torus.

The area surrounding a knot is a knotted full torus.

Clear description

The 3-dimensional sphere is the sphere that arises from the 3-dimensional space by adding a point at infinity , in short the one-point compactification of the 3-dimensional space. A node is a circular line embedded in the 3-dimensional sphere. A neighborhood of this node is a full torus , the edge of this neighborhood is a torus .

By cutting out this full torus from the 3-dimensional sphere, a 3-dimensional manifold is obtained , the edge of which is a torus. (See Node Complement .)

By means of a glued image, which is a self-image of the torus, the full torus can now be glued to the edge again and a closed three-dimensional manifold is obtained.

This new 3-manifold generally has a different topology than the 3-sphere, namely precisely when the adhesive mapping is not homotopic to the identity mapping .

Correspondingly, you can cut out a surrounding area for nodes embedded in other 3-manifolds and glue them back in differently. This procedure is known as stretching surgery.

Mathematical definition

Let be a 3-manifold and an embedding with image . Let be an integer matrix. Man folders on to by using identified. ${\displaystyle M}$ ${\displaystyle t\colon \mathbb {D} ^{2}\times S^{1}\rightarrow M}$ ${\displaystyle U}$ ${\displaystyle ({\begin{array}{cc}a&b\\c&d\end{array}})\in SL(2,\mathbb {Z} )}$ ${\displaystyle S^{1}\times \mathbb {D} ^{2}}$ ${\displaystyle M\setminus U}$ ${\displaystyle (z,w)\in S^{1}\times S^{1}\subset S^{1}\times \mathbb {D} ^{2}}$ ${\displaystyle t(z^{a}w^{b},z^{c}w^{d})}$

One can show that the manifold constructed in this way only depends on the node and the numbers (not on ), except for homeomorphism . It is called the manifold obtained by stretching the knot with coefficients . ${\displaystyle M_{K}(-{\tfrac {b}{d}})}$ ${\displaystyle K=t(0\times S^{1})}$ ${\displaystyle b,d}$ ${\displaystyle a,c}$ ${\displaystyle M_{K}(-{\tfrac {b}{d}})}$ ${\displaystyle K}$ ${\displaystyle -{\tfrac {b}{d}}}$

Accordingly, there is an entanglement (Link) a manifold by sequential execution of coefficients (in any order) of Dehn surgeries at the nodes define. ${\displaystyle L=K_{1}\cup \ldots \cup K_{n}\subset M}$ ${\displaystyle M_{L}(-{\tfrac {b_{1}}{d_{1}}},\ldots ,{\tfrac {b_{n}}{d_{n}}})}$ ${\displaystyle -{\tfrac {b_{i}}{d_{i}}}}$ ${\displaystyle K_{i},i=1\ldots ,n}$

Construction of 3-Manifolds (Lickorish-Wallace Theorem)

Every closed, orientable, connected 3-manifold can be constructed by stretching surgery on a link in the 3-sphere. One can even achieve that all components of are untied and that all are coefficients . ${\displaystyle L\subset S^{3}}$ ${\displaystyle L}$ ${\displaystyle -{\tfrac {b_{i}}{d_{i}}}=\pm 1}$

Construction of hyperbolic 3-manifolds (Thurston's theorem)

If a complete hyperbolic metric carries finite volume, then almost all of the manifolds generated by stretching surgery on are also hyperbolic. ${\displaystyle M\setminus K}$ ${\displaystyle K}$

There are 10 exceptional (i.e. non-hyperbolic) stretching surgeries for the figure eight knot . Lackenby and Meyerhoff have shown that for each knot the number of exceptional stretching surgeries is at most 10.

See also

• Property P
• Property R

Web links

• Siddhartha Gadgil, Dehn Surgery, pdf

literature

• Max Dehn : About the topology of three-dimensional space. Math. Ann. 1910, 69: 137-168.
• Chapter 65 in: Herbert Seifert , William Threlfall : Textbook of Topology , 89, Leipzig: Teubner (1934).

supporting documents

1. ↑ tom Dieck, Tammo: Algebraic topology. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zurich, 2008. ISBN 978-3-03719-048-7
2. ^ Wallace, Andrew H .: Modifications and cobounding manifolds. Canad. J. Math. 12 1960 503-528.
3. ↑ Lickorish, WBR: A representation of orientable combinatorial 3-manifolds. Ann. of Math. (2) 76 1962 531-540.
4. ^ Thurston, WP: The Geometry and Topology of Three-Manifolds
5. ↑ Lackenby, Marc; Meyerhoff, Robert: The maximal number of exceptional stretching surgeries. Invent. Math. 191 (2013), no.2, 341-382. pdf