Algebraic surgery theory

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 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 interesting invariants of the manifold are known.

The classification of exotic spheres by Kervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.

If X, Y are manifolds with boundary, then the boundary of the product manifold is ∂(X × Y)=(∂X × Y) ∪ (X × ∂Y). The basic observation which justifies surgery is that the space S^p × S^q−1 can be understood either as the boundary of D^p+1 × S^q-1 or as the boundary of S^p × D^q. In symbols, ∂(S^p × D^q) = S^p × S^q−1 = ∂(D^p+1 × S^q−1), where D^q is the q-dimensional disk, i.e., the set of q-dimensional points that are at distance one-or-less from a given fixed point (the center of the disk); for example, then, D¹ is (equivalent, or homeomorphic to), the unit interval, while D² is a circle together with the points in its interior.

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