Dehn surgery

Definitions

• Given a 3-manifold ${\displaystyle M}$ and a link ${\displaystyle L\subset M}$, the manifold ${\displaystyle M}$ drilled along ${\displaystyle L}$ is obtained by removing an open tubular neighborhood of ${\displaystyle L}$ from ${\displaystyle M}$. If ${\displaystyle L=L_1\cup \cdots \cup L_k}$, the drilled manifold has ${\displaystyle k}$ torus boundary components ${\displaystyle T_1\cup \cdots \cup T_k}$. The manifold ${\displaystyle M}$ drilled along ${\displaystyle L}$ is also known as the link complement, since if one removed the corresponding closed tubular neighborhood from ${\displaystyle M}$, one obtains a manifold diffeomorphic to ${\displaystyle M\setminus L}$.
• Given a 3-manifold whose boundary is made of 2-tori ${\displaystyle T_1\cup \cdots \cup T_k}$, we may glue in one solid torus by a homeomorphism (resp. diffeomorphism) of its boundary to each of the torus boundary components ${\displaystyle T_i}$ of the original 3-manifold. There are many inequivalent ways of doing this, in general. This process is called Dehn filling.
• Dehn surgery on a 3-manifold containing a link consists of drilling out a tubular neighbourhood of the link together with Dehn filling on all the components of the boundary corresponding to the link.

In order to describe a Dehn surgery, one picks two oriented simple closed curves ${\displaystyle m_i}$ and ${\displaystyle {\ell }_i}$ on the corresponding boundary torus ${\displaystyle T_i}$ of the drilled 3-manifold, where ${\displaystyle m_i}$ is a meridian of ${\displaystyle L_i}$ (a curve staying in a small ball in ${\displaystyle M}$ and having linking number +1 with ${\displaystyle L_i}$ or, equivalently, a curve that bounds a disc that intersects the component ${\displaystyle L_i}$ once) and ${\displaystyle {\ell }_i}$ is a longitude of ${\displaystyle T_i}$ (a curve travelling once along ${\displaystyle L_i}$ or, equivalently, a curve on ${\displaystyle T_i}$ such that the algebraic intersection ${\displaystyle ⟨{\ell }_i,m_i⟩}$ is equal to +1). The curves ${\displaystyle m_i}$ and ${\displaystyle {\ell }_i}$ generate the fundamental group of the torus ${\displaystyle T_i}$, and they form a basis of its first homology group. This gives any simple closed curve ${\displaystyle {\gamma }_i}$ on the torus ${\displaystyle T_i}$ two coordinates ${\displaystyle a_i}$ and ${\displaystyle b_i}$, so that ${\displaystyle [{\gamma }_i]=[a_i{\ell }_i+b_i{m}_i]}$. These coordinates only depend on the homotopy class of ${\displaystyle {\gamma }_i}$.

We can specify a homeomorphism of the boundary of a solid torus to ${\displaystyle T_i}$ by having the meridian curve of the solid torus map to a curve homotopic to ${\displaystyle {\gamma }_i}$. As long as the meridian maps to the surgery slope ${\displaystyle [{\gamma }_i]}$, the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism). The ratio ${\displaystyle b_i/a_i\in \mathbb{Q}\cup \{\mathrm{\infty }\}}$ is called the surgery coefficient of ${\displaystyle L_i}$.

In the case of links in the 3-sphere or more generally an oriented integral homology sphere, there is a canonical choice of the longitudes ${\displaystyle {\ell }_i}$: every longitude is chosen so that it is null-homologous in the knot complement—equivalently, if it is the boundary of a Seifert surface.

When the ratios ${\displaystyle b_i/a_i}$ are all integers (note that this condition does not depend on the choice of the longitudes, since it corresponds to the new meridians intersecting the old meridians exactly once), the surgery is called an integral surgery. Such surgeries are closely related to handlebodies, cobordism and Morse functions.