Surgery theory

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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 on a manifold[edit]


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

M':= (M-\operatorname{int~im}\phi)\;\cup_{\phi|_{S^p\times S^{q-1}}} (D^{p+1} \!\times\! S^{q-1}).


Surgery is closely related to (but not the same as) handle attaching. Given an (n+1)-manifold with boundary (L, ∂L) and an embedding \phi: Sp × Dq → ∂L, where n = p+q, define another (n+1)-manifold with boundary L′ by

L':= L\; \cup_\phi (D^{p+1}\!\!\times\!D^q).




 W:= (M\times I)\; \cup_{S^p\times D^q\times \{1\}} (D^{p+1}\!\times\!D^q)





1. Surgery on the circle

Fig. 1


Fig. 2a
Fig. 2b

2. Surgery on the 2-sphere



3. Surgery on the n-sphere

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 Sn 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.

4. Morse functions

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+\epsilon) is obtained from M:=f^{-1}(c-\epsilon) by a p-surgery. The bordism W:=f^{-1}([c-\epsilon,c+\epsilon]) 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 × 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.


Effects on homotopy groups, and comparison to cell-attachment[edit]



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 φ: Sp × DqM (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[edit]


  • Is X a manifold?
  • Is f a diffeomorphism?

More formally, one must ask whether up to homotopy:




The surgery approach[edit]

In the classical approach, as developed by Browder, Novikov, Sullivan and Wall, surgery is done on normal maps of degree one. Using surgery, the question "Is the normal map f: MX of degree one cobordant to a homotopy equivalence?" can be translated (in dimensions greater than four) to an algebraic statement about some element in an L-group of the group ring \mathbf{Z}[\pi_1(X)]. More precisely, the question has a positive answer if and only if the surgery obstruction \sigma(f)\in L_n(\mathbf{Z}[\pi_1(X)]) is zero, where n is the dimension of M.

For example, consider the case where the dimension n=4k is a multiple of four, and \pi_1(X)=0. It is known that L_{4k}(\mathbf{Z}) is isomorphic to the integers \mathbf{Z}; under this isomorphism the surgery obstruction of f maps, up to a scalar factor, to the difference of the signatures \sigma(X)-\sigma(M) of X and M. Hence a normal map of degree one is cobordant to a homotopy equivalence if and only if the signatures of domain and codomain agree.


[X, G/O] \to L_n(\mathbf{Z}[\pi_1(X)]).

Structure sets and surgery exact sequence[edit]




See also[edit]


External links[edit]