Төрт өлшемді топологиялық кеңістіктерді өзгерту операциясы

Operation used to modify three dimensional topological spaces
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3 manifolds. The process takes as input a 3 manifold together with a link. It is often conceptualized as two steps: drilling then filling.

Given a 3 manifold and a link , the manifold drilled along is obtained by removing an open tubular neighborhood of from If , the drilled manifold has torus boundary components The manifold drilled along is also known as the link complement, since if one removed the corresponding closed tubular neighborhood from , one obtains a manifold diffeomorphic to Given a 3 manifold whose boundary is made of 2 tori , we may glue in one solid torus by a homeomorphism (resp. diffeomorphism) of its boundary to each of the torus boundary components 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 and on the corresponding boundary torus of the drilled 3 manifold, where is a meridian of (a curve staying in a small ball in and having linking number +1 with or, equivalently, a curve that bounds a disc that intersects once the component ) and is a longitude of (a curve travelling once along or, equivalently, a curve on such that the algebraic intersection is equal to +1). The curves and generate the fundamental group of the torus , and they form a basis of its first homology group. This gives any simple closed curve on the torus two coordinates and , so that These coordinates only depend on the homotopy class of
We can specify a homeomorphism of the boundary of a solid torus to by having the meridian curve of the solid torus map to a curve homotopic to As long as the meridian maps to the surgery slope , the resulting Dehn surgery will yield a 3 manifold that will not depend on the specific gluing (up to homeomorphism). The ratio is called the surgery coefficient of
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 : 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 are all integers (note that this condition does not depend on the choice of the longitudes, since it corresponds to the new meridians intersecting exactly once the ancient meridians), the surgery is called an integral surgery. Such surgeries are closely related to handlebodies, cobordism and Morse functions.

If all surgery coefficients are infinite, then each new meridian is homotopic to the ancient meridian Therefore the homeomorphism type of the manifold is unchanged by the surgery. If is the 3 sphere, is the unknot, and the surgery coefficient is , then the surgered 3 manifold is
If is the 3 sphere, is the unknot, and the surgery coefficient is , then the surgered 3 manifold is the lens space In particular if the surgery coefficient is of the form , then the surgered 3 manifold is still the 3 sphere. If is the 3 sphere, is the right handed trefoil knot, and the surgery coefficient is , then the surgered 3 manifold is the Poincaré dodecahedral space.

Every closed, orientable, connected 3 manifold is obtained by performing Dehn surgery on a link in the 3 sphere. This result, the Lickorish–Wallace theorem, was first proven by Andrew H. Wallace in 1960 and independently by W. B. R. Lickorish in a stronger form in 1962. Via the now well known relation between genuine surgery and cobordism, this result is equivalent to the theorem that the oriented cobordism group of 3 manifolds is trivial, a theorem originally proved by Vladimir Abramovich Rokhlin in 1951. Since orientable 3 manifolds can all be generated by suitably decorated links, one might ask how distinct surgery presentations of a given 3 manifold might be related. The answer is called the Kirby calculus.