Surgery theory | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Surgery_theory


00:01:28 1 Surgery on a manifold
00:02:10 1.1 Examples
00:08:50 1.2 Effects on homotopy groups, and comparison to cell-attachment
00:17:37 2 Application to classification of manifolds
00:20:02 2.1 The surgery approach
00:21:33 2.2 Structure sets and surgery exact sequence
00:26:13 3 See also
00:28:56 4 References



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SUMMARY
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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). Originally developed for  differentiable (= smooth) manifolds, surgery techniques also apply to PL (= piecewise linear) 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. It is a major tool in the study and classification of manifolds of dimension greater than 3.
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.
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.