A basic observation If
X,
Y are manifolds with boundary, then the boundary of the product manifold is :\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 \times S^{q-1} can be understood either as the boundary of D^{p+1} \times S^{q-1} or as the boundary of S^p \times D^q. In symbols, : \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 is the
q-dimensional disk, i.e., the set of points in \R^q that are at distance one-or-less from a given fixed point (the center of the disk); for example, then, D^1 is
homeomorphic to the unit interval, while D^2 is a circle together with the points in its interior.
Surgery Now, given a manifold
M of dimension n = p+q and an
embedding \phi\colon S^p \times D^q \to M, define another
n-dimensional manifold M' to be :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). Since \operatorname{im}(\phi)=\phi(S^p \times D^q) and from the equation from our basic observation before, the gluing is justified then :\phi\left(\partial\left(S^p \times D^q\right)\right) = \phi\left(S^p \times S^{q-1}\right). One says that the manifold
M′ is produced by a
surgery cutting out S^p \times D^q and gluing in D^{p+1} \times S^{q-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).
Attaching handles and cobordisms Surgery is closely related to (but not the same as)
handle attaching. Given an (n+1)-manifold with boundary (L, \partial L) and an embedding \phi\colon S^p\times D^q \to \partial L, where n = p+q, define another (n+1)-manifold with boundary
L′ by :L' := L\; \cup_\phi \left(D^{p+1} \times D^q\right). The manifold
L′ is obtained by "attaching a (p+1)-handle", with \partial L' obtained from \partial L by a
p-surgery :\partial L' = (\partial L \setminus \operatorname{int} ( \operatorname{im}(\phi)) ) \; \cup_{\phi|_{S^p \times D^{q}}} \left(D^{p+1} \times S^{q-1}\right). 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 :W := (M \times I)\; \cup_{\phi \times \{1\}} \left(D^{p+1} \times D^q\right) the (n+1)-dimensional manifold with boundary \partial W = M\cup M' obtained from the product M\times I by attaching a (p+1)-handle D^{p+1} \times D^q. 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 {{ordered list As per the above definition, a surgery on the circle consists of cutting out a copy of S^0\times D^1 and gluing in D^1\times S^0. The pictures in Fig. 1 show that the result of doing this is either (i) S^1 again, or (ii) two copies of S^1. In this case there are more possibilities, since we can start by cutting out either S^1\times D^1 or S^0\times D^2. If n=p+q, then :S^n=\partial D^{n+1}\approx \partial (D^{p+1}\times D^q) = S^p\times D^q\;\cup\;D^{p+1}\times S^{q-1}. The
p-surgery on 'S^n is therefore :D^{p+1}\times S^{q-1}\;\cup\;D^{p+1}\times S^{q-1} = S^{p+1}\times S^{q-1}. Examples 1 and 2 above were a special case of this. Suppose that
f is a
Morse function on an (
n + 1)-dimensional manifold, and suppose that
c is a critical value with exactly one critical point in its preimage. If the index of this critical point is p+1, then the level-set M':= f^{-1}(c + \varepsilon) is obtained from M:= f^{-1}(c - \varepsilon) by a
p-surgery. The bordism W:= f^{-1}([c - \varepsilon, c + \varepsilon]) can be identified with the trace of this surgery. Indeed, in some coordinate chart around the critical point, the function
f is of the form -\Vert x\Vert^2 + \Vert y\Vert^2, with x\in R^{p+1}, y\in R^q, and p+q+1 =n+1. Fig. 3 shows, in this local chart, the manifold
M in blue and the manifold
M′ in red. The colored region between
M and
M′ corresponds to the bordism
W. The picture shows that
W is diffeomorphic to the union :W \cong M \times I \cup_{S^p\times D^q} D^{p+1}\times D^q (neglecting the issue of straightening corners), where M\times I is colored in yellow, and D^{p+1}\times D^q is colored in green. The manifold
M′, being a boundary component of
W, is therefore obtained from
M by a
p-surgery. Since every bordism between closed manifolds has a Morse function where different critical points have different critical values, this shows that any bordism can be decomposed into traces of surgeries (
handlebody decomposition). In particular, every manifold
M may be regarded as a bordism from the boundary ∂
M (which may be empty) to the empty manifold, and so may be obtained from \partial M\times I by attaching handles. }}
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 \phi 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 \phi\colon S^p\times D^q \to 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. In the same way as a cell can be attached to a space to kill an element in some
homotopy group of the space, a
p-surgery on a manifold
M can often be used to kill an element \alpha\in\pi_p(M). Two points are important however: Firstly, the element \alpha\in\pi_p(M) has to be representable by an embedding \phi\colon S^p\times D^q \to M (which means embedding the corresponding sphere with a trivial
normal bundle). For instance, it is not possible to perform surgery on an orientation-reversing loop. Secondly, the effect of the detaching process has to be considered, since it might also have an effect on the homotopy group under consideration. Roughly speaking, this second point is only important when
p is at least of the order of half the dimension of
M. ==Application to classification of manifolds==