nLab
Dehn surgery

Contents

Context

Manifolds and cobordisms

Knot theory

Contents

Idea

Dehn surgery is a method for constructing one manifold from another, especially one 3-manifold from another, by a kind of cut-and-paste procedure.

Method

A standard application of Dehn surgery is surgery along a link LL in a 3-sphere S 3S^3. This works in two steps (whose description makes it sound like Zahn surgery1):

  • Remove or excise from S 3S^3 a tubular neighborhood of the embedded link LS 3L \hookrightarrow S^3, of the form L×int(D 2)L \times int(D^2). This is called Dehn drilling. The result is a 3-manifold with boundary MM, whose boundary ML×S 1\partial M \cong L \times S^1 can be viewed as the boundary of a disjoint union of solid tori L×D 2L \times D^2.

  • To each of the components C 1,,C nC_1, \ldots, C_n of M\partial M, apply an (orientation-preserving) homeomorphism, say ϕ 1,,ϕ n\phi_1, \ldots, \phi_n. The union ϕ 1ϕ n\phi_1 \cup \ldots \cup \phi_n is a homeomorphism ϕ:MM\phi: \partial M \to \partial M. Then perform a Dehn filling by constructing the pushout of an evident span:

    M ϕ M incl incl M L×D 2,\array{ \partial M & \stackrel{\phi}{\to} & \partial M \\ \mathllap{incl} \downarrow & & \downarrow \mathrlap{incl} \\ M & & L \times D^2, }

    thus refilling the drilled portion, but in a new way (along ϕ\phi). This gives a new 3-manifold NN.

Some further notes: the surgery can be done one solid torus at a time. A homeomorphism on a boundary torus S 1×S 1S^1 \times S^1 sends a meridian S 1×{1}S_1 \times \{1\} to some simple closed curve that is homotopic to a curve of rational slope p/qp/q (the curve which is the image of the line y=(p/q)xy = (p/q)x in 2\mathbb{R}^2 under the quotient map 2 2/ 2S 1×S 1\mathbb{R}^2 \to \mathbb{R}^2/\mathbb{Z}^2 \cong S^1 \times S^1). It turns out that the result of the surgery depends, up to homeomorphism, only on the quantity p/qp/q, called a surgery coefficient. If all the surgery coefficients are integers, we speak of an integral surgery.

Put a bit different: given a framed link in an oriented 3-manifold like S 3S^3, an integral surgery drills out a solid torus, twists it an integral number of times according to the framing, and then reattaches it.

Results

Theorem

(Lickorish-Wallace) Every connected oriented closed 3-manifold NN arises by performing an integral Dehn surgery along a link LS 3L \hookrightarrow S^3 (i.e., surgery along a framed link).

References

See also


  1. Yes, that’s supposed to be a little joke. ‘Zahn’ here is the German word.

Last revised on December 30, 2018 at 12:57:57. See the history of this page for a list of all contributions to it.