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. Examples 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. Results 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. See also Hyperbolic Dehn surgery Tubular neighborhood Surgery on manifolds, in the general sense, also called spherical modification. References . . . . 3-manifolds Surgery theory