Surgery Theory - Surgery On A Manifold

Surgery On A Manifold

Recall that in general, 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 Sp × Sq-1 can be understood either as the boundary of Dp+1 × Sq-1 or as the boundary of Sp × Dq. In symbols, ∂(Sp × Dq) = Sp × Sq-1 = ∂(Dp+1 × Sq-1), where Dq 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.

Now, given a manifold M of dimension n = p+q and an embedding : Sp × Dq → M, define another n-dimensional manifold M′ to be

One says that the manifold M′ is produced by a surgery cutting out Sp × Dq and gluing in Dp+1 × Sq-1, 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).

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.