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In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961). Milnor called this technique surgery, while Andrew Wallace called it spherical modification.[1] The "surgery" on a differentiable manifold M of dimension , could be described as removing an imbedded sphere of dimension p from M.[2] Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.
Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions.
More technically, the idea is to start with a well-understood manifold M and perform surgery on it to produce a manifold M′ having some desired property, in such a way that the effects on the homology, homotopy groups, or other invariants of the manifold are known. A relatively easy argument using Morse theory shows that a manifold can be obtained from another one by a sequence of spherical modifications if and only if those two belong to the same cobordism class.[1]
The classification of exotic spheres by Michel Kervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.
In mathematics, specifically in geometric topology, surgerytheory is a collection of techniques used to produce one finite-dimensional manifold from another...
manifolds and differentiable manifolds coincide. The two basic questions of surgerytheory are whether a topological space with n-dimensional Poincaré duality...
Singularity theory Soliton theory Spectral theory String theory Sturm-Liouville theorySurgerytheory Teichmüller theoryTheory of equations Theory of statistics...
structure; high-dimensional manifolds are classified algebraically, by surgerytheory. "Low dimensions" means dimensions up to 4; "high dimensions" means...
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...
Set theory — Shape theory — Small cancellation theory — Spectral theory — Stability theory — Stable theory — Sturm–Liouville theory — Surgerytheory — Twistor...
Cappell, Sylvain; Ranicki, Andrew; Rosenberg, Jonathan (eds.), Surveys on surgerytheory. Vol. 1, Annals of Mathematics Studies, Princeton University Press,...
topology due to William Browder which is of fundamental importance in surgerytheory. Given a Poincaré complex X (more geometrically a Poincaré space), a...
Orr, K (2000), "A survey of applications of surgery to knot and link theory", Surveys on SurgeryTheory: Papers Dedicated to C.T.C. Wall, Annals of mathematics...
characteristic 2 (applied in knot theory and surgerytheory) in topology, the Hasse–Arf theorem in ramification theory, Arf semigroups and Arf rings. Cahit...
pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgerytheory of high-dimensional manifolds. Handles are used to...
theory, and h-cobordisms are fundamental in the study of high-dimensional manifolds, namely surgerytheory. In algebraic topology, cobordism theories...
can be analyzed by surgerytheory similarly to manifolds, and failure to be a manifold is a local obstruction, as in surgerytheory. Differential spaces...
Bariatric surgery (or metabolic surgery or weight loss surgery) is a medical term for surgical procedures used to manage obesity and obesity-related conditions...
Normal mapping in 3D computer graphics Normal invariants in mathematical surgerytheory Normal matrix in linear algebra Normal operator in functional analysis...
Oral and maxillofacial surgery is a surgical specialty focusing on reconstructive surgery of the face, facial trauma surgery, the oral cavity (mouth)...
J. H. C. Whitehead. The Whitehead torsion is important in applying surgerytheory to non-simply connected manifolds of dimension > 4: for simply-connected...
smoothable topological constructions, such as smooth surgerytheory or the construction of cobordisms. Morse theory is an important tool which studies smooth manifolds...
International Journal of Surgery: Devoted to the Theory and Practice of Modern Surgery and Gynecology. The International Journal of Surgery Co. 1919. p. 392....
topology, characteristic classes are a basic invariant, and surgerytheory is a key theory. Low-dimensional topology is strongly geometric, as reflected...
in the development of surgerytheory. In fact, these calculations can be formulated in a modern language in terms of the surgery exact sequence as indicated...
"Refractive surgery". Theory and practice of optics and refraction (2nd ed.). Elsevier. pp. 307–348. ISBN 978-81-312-1132-8. "Laser Eye Surgery". MedlinePlus...
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