manifolds and cobordisms
cobordism theory, Introduction
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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.
A standard application of Dehn surgery is surgery along a link $L$ in a 3-sphere $S^3$. This works in two steps (whose description makes it sound like Zahn surgery^{1}):
Form the complement in the 3-sphere of a tubular neighborhood of the embedded link $L \hookrightarrow S^3$, of the form $L \times int(D^2)$. This is called Dehn drilling. The result is a 3-manifold with boundary $M$, whose boundary $\partial M \cong L \times S^1$ can be viewed as the boundary of a disjoint union of solid tori $L \times D^2$.
To each of the connected components $C_1, \ldots, C_n$ of $\partial M$, apply an (orientation-preserving) homeomorphism, say $\phi_1, \ldots, \phi_n$. The union $\phi_1 \cup \ldots \cup \phi_n$ is a homeomorphism $\phi: \partial M \to \partial M$. Then perform a Dehn filling by constructing the pushout of an evident span:
thus refilling the drilled portion, but in a new way (along $\phi$). This gives a new 3-manifold $N$.
Some further notes: the surgery can be done one solid torus at a time. A homeomorphism on a boundary torus $S^1 \times S^1$ sends a meridian $S_1 \times \{1\}$ to some simple closed curve that is homotopic to a curve of rational slope $p/q$ (the curve which is the image of the line $y = (p/q)x$ in $\mathbb{R}^2$ under the quotient map $\mathbb{R}^2 \to \mathbb{R}^2/\mathbb{Z}^2 \cong S^1 \times S^1$). It turns out that the result of the surgery depends, up to homeomorphism, only on the quantity $p/q$, 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 $S^3$, an integral surgery drills out a solid torus, twists it an integral number of times according to the framing, and then reattaches it.
(Lickorish-Wallace) Every connected oriented closed 3-manifold $N$ arises by performing an integral Dehn surgery along a link $L \hookrightarrow S^3$ (i.e., surgery along a framed link).
See also
Relation between Dehn surgery and Wilson loop observables in Chern-Simons theory:
E. Guadagnini, Surgery rules in quantum Chern-Simons field theory, Nuclear Physics B Volume 375, Issue 2, 18 May 1992, Pages 381-398 (doi:10.1016/0550-3213(92)90037-C)
Boguslaw Broda, Chern-Simons theory on an arbitrary manifold via surgery (arXiv:hep-th/9305051)
Yes, that’s supposed to be a little joke. ‘Zahn’ here is the German word. ↩
Last revised on December 18, 2019 at 23:08:41. See the history of this page for a list of all contributions to it.