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 called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension , 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 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
The basic observation which justifies surgery is that the space can be understood either as the boundary of or as the boundary of . In symbols, ,
where is the q-dimensional disk, i.e., the set of points in that are at distance one-or-less from a given fixed point (the center of the disk); for example, then, is homeomorphic to the unit interval, while is a circle together with the points in its interior.
Surgery
Now, given a manifold M of dimension and an embedding , define another n-dimensional manifold to be
Since and from the equation from our basic observation before, the gluing is justified then
One says that the manifold M′ is produced by a surgery cutting out and gluing in , or by a p-surgery if one wants to specify the number p. Strictly speaking, M′ is a manifold with corners, but there is a canonical way to smooth them out. Notice that the submanifold that was replaced in M was of the same dimension as M (it was of codimension 0).
Attaching handles and cobordisms
Surgery is closely related to (but not the same as) handle attaching. Given an (n + 1)-manifold with boundary (L, ∂L) and an embedding : Sp × Dq → ∂L, where n = p + q, define another (n + 1)-manifold with boundary L′ by
The manifold L′ is obtained by "attaching a (p + 1)-handle", with ∂L′ obtained from ∂L by a p-surgery
A surgery on M not only produces a new manifold M′, but also a cobordism W between M and M′. The trace of the surgery is the cobordism (W; M, M′), with
the (n + 1)-dimensional manifold with boundary ∂W = M ∪ M′ obtained from the product M × I by attaching a (p + 1)-handle Dp+1 × Dq.
Surgery is symmetric in the sense that the manifold M can be re-obtained from M′ by a (q − 1)-surgery, the trace of which coincides with the trace of the original surgery, up to orientation.
In most applications, the manifold M comes with additional geometric structure, such as a map to some reference space, or additional bundle data. One then wants the surgery process to endow M′ with the same kind of additional structure. For instance, a standard tool in surgery theory is surgery on normal maps: such a process changes a normal map to another normal map within the same bordism class.
Examples
Effects on homotopy groups, and comparison to cell-attachment
Intuitively, the process of surgery is the manifold analog of attaching a cell to a topological space, where the embedding φ takes the place of the attaching map. A simple attachment of a (p + 1)-cell to an n-manifold would destroy the manifold structure for dimension reasons, so it has to be thickened by crossing with another cell.
Up to homotopy, the process of surgery on an embedding φ: Sp × Dq → M can be described as the attaching of a (p + 1)-cell, giving the homotopy type of the trace, and the detaching of a q-cell to obtain N. The necessity of the detaching process can be understood as an effect of Poincaré duality.