Dehn surgery

Let be a closed -dimensional manifold and let be a solid torus in the interior of . Remove from and glue in instead of it another solid torus by a homeomorphism . One says that the resulting new -dimensional manifold

is obtained from by a Dehn surgery. Note that is determined by the following data: 1) a knot (a core circle of the solid torus , cf. also Knot theory); and 2) a non-trivial simple closed curve (the image under of a meridian of ). The Dehn surgery is called integer if is a longitude of , i.e., intersects a meridional curve of transversally in a single point.

If , then among all longitudes of there is a preferred one, which bounds a surface in the complement of . The preferred longitude forms together with a meridian of a coordinate system on . Therefore, has the form , where are coprime integers, and is determined by the rational number . The Dehn surgery is integer if and only if is an integer. This explains the terminology.

Let be two handle-bodies having the same genus (cf. Handle theory) and let be a homeomorphism. Denote by the closed -dimensional manifold obtained by gluing and along . Choose a simple closed curve and denote by the Dehn twist along . To be more precise, is a homeomorphism obtained by cutting along , isotopically rotating one side of the cut by , and gluing back. Let . Since and coincide outside a neighbourhood of in , and do actually coincide outside regular neighbourhoods of in and , respectively. It follows that is obtained from by a Dehn surgery along . One can easily show that the surgery is integer.

Define a framed link to be a link such that every component of is supplied with an integer number , called a framing. If one performs Dehn surgeries along all components of , taking for each component the framing as the parameter of the surgery, one obtains a -dimensional manifold . Since any orientation-preserving homeomorphism of the boundary of a handle-body is isotopic to a product of Dehn twists [a1], it follows from the above relation between Dehn twists and integer Dehn surgeries that for every closed orientable -dimensional manifold there exists a framed link such that .

The following question naturally arises: When do two framed links determine homeomorphic -dimensional manifolds? In 1978 R. Kirby answered this question by showing that if and only if one can pass from to by a sequence of the following moves and their inverses [a2]:

1) replace by the link , where is a new unknotted component with framing such that is contained in a -dimensional ball , ;

2) replace a component by a geometric sum , , of with another component (see [a2] for the exact definition of the geometric sum).

This result became broadly known as the Kirby calculus for framed links, thanks to its convenience for presenting -dimensional manifolds.

Recall that a framing of a knot determines a homeomorphism of the standard solid torus onto a regular neighbourhood of . Denote by the -dimensional manifold obtained by attaching a -dimensional handle of index to the -dimensional ball via the homeomorphism between and . It follows from the definition that . Similarly, for any framed link the -dimensional manifold is the boundary of the -dimensional manifold obtained by attaching handles of index to the -dimensional ball. Move 1) on corresponds to replacing by a connected sum of with . Move 2) corresponds to a sliding of one handle of index over another and does not change . One can show that any framed link in can be transformed by moves 1), 2) and their inverses to a link with even framings [a3]. In the latter case the tangent bundle of the corresponding -dimensional manifold is trivial.

See [a4] for more details.

References

[a1]	W.B.R. Lickorish, "A representation of orientable combinatorial 3-manifolds" Ann. Math , 76 (1962) pp. 531–540
[a2]	R. Kirby, "A calculus for framed links in " Invent. Math. , 45 (1978) pp. 35–56
[a3]	S. Kaplan, "Constructing framed 4-manifolds with given almost framed boundaries" Trans. Amer. Math. Soc. , 254 (1979) pp. 237–263
[a4]	A.T. Fomenko, S.V. Matveev, "Algorithmic and computer methods in three dimensional topology" , Kluwer Acad. Publ. (1997)