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. The "surgery" on a differentiable manifold M of dimension n = p + q + 1 {\displaystyle n=p+q+1} , could be described as removing an imbedded sphere of dimension p from M. 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. 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.
Surgery on a manifold
A basic observation If X, Y are manifolds with boundary, then the boundary of the product manifold is
∂ ( X × Y ) = ( ∂ X × Y ) ∪ ( X × ∂ Y ) . {\displaystyle \partial (X\times Y)=(\partial X\times Y)\cup (X\times \partial Y).}
The basic observation which justifies surgery is that the space S p × S q − 1 {\displaystyle S^{p}\times S^{q-1}} can be understood either as the boundary of D p + 1 × S q − 1 {\displaystyle D^{p+1}\times S^{q-1}} or as the boundary of S p × D q {\displaystyle S^{p}\times D^{q}} . In symbols,
∂ ( S p × D q ) = S p × S q − 1 = ∂ ( D p + 1 × S q − 1 ) {\displaystyle \partial \left(S^{p}\times D^{q}\right)=S^{p}\times S^{q-1}=\partial \left(D^{p+1}\times S^{q-1}\right)} , where D q {\displaystyle D^{q}} is the q-dimensional disk, i.e., the set of points in R q {\displaystyle \mathbb {R} ^{q}} that are at distance one-or-less from a given fixed point (the center of the disk); for example, then, D 1 {\displaystyle D^{1}} is homeomorphic to the unit interval, while D 2 {\displaystyle D^{2}} is a circle together with the points in its interior.
Surgery Now, given a manifold M of dimension n = p + q {\displaystyle n=p+q} and an embedding ϕ : S p × D q → M {\displaystyle \phi \colon S^{p}\times D^{q}\to M} , define another n-dimensional manifold M ′ {\displaystyle M'} to be
M ′ := ( M ∖ int ( im ( ϕ ) ) ) ∪ ϕ | S p × S q − 1 ( D p + 1 × S q − 1 ) . {\displaystyle M':=\left(M\setminus \operatorname {int} (\operatorname {im} (\phi ))\right)\;\cup _{\phi |_{S^{p}\times S^{q-1}}}\left(D^{p+1}\times S^{q-1}\right).}
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