Surgery theory

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In mathematics, specifically in topology, surgery theory is the name given to a collection of techniques used to produce one manifold from another in a 'controlled' way. 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.

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 interesting invariants of the manifold are known.

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[edit] Definition

The intuitive idea of surgery is simple: it is the manifold analog of attaching a cell to a topological space. A surgery on an n-dimensional manifold with n = p + q removes S^p\times D^q from M and replaces it by D^{p+1}\times S^{q-1} to create a new manifold N= (M - S^p \times D^q) \cup (D^{p+1}\times S^{q-1}) . The trace of the surgery is the cobordism (Wn + 1;M,N) with W = M \times I \cup_{S^p \times D^q \times \{1\}} D^{p+1} \times D^q. The homotopy-theoretic effect of a surgery is to attach a (p + 1)-cell to M, giving the homotopy type of W, and then to detach the complementary (q − 1)-cell to obtain N.

In the theory of CW complexes, attaching a k-cell doesn't affect the topology below dimension k − 1, so you can build spaces up by attaching cells in successively higher dimensions. You cannot naively do this to manifolds: manifolds have the structure of a CW complex with Poincaré duality, but you cannot attach a single k-cell to an n-dimensional manifold and have it stay a manifold, because you've glued a k-dimensional object onto an n-dimensional one. Instead, you must "thicken" the cell up to being n-dimensional.

In CW complexes, given a class \alpha \in \pi_p X, you can kill it by attaching a p + 1 cell to X via an attaching map f:S^p\to X that represents the class α.

Points to note:

  • Given an embedded sphere S^p \subset M, with trivial normal bundle, a tubular neighborhood of it is diffeomorphic to S^p \times D^q. This is a manifold with boundary S^p \times S^{q-1}, which is also the boundary of D^{p+1} \times S^{q-1}.
  • To perform surgery on a homotopy/homology class, it must be representable by an embedded sphere with trivial normal bundle. (For instance, you cannot perform surgery on an orientation-reversing loop.)
  • If M=\partial W is the boundary of an (n + 1)-dimensional manifold W, then W'=W \cup_{S^p \times D^q}D^{p+1} \times D^{q+1} is the (n + 1)-dimensional manifold obtained from W by 'attaching a (p + 1)-handle', with boundary \partial W=N obtained from M by surgery.
  • The surgered manifold has a PDIFF (= piecewise smooth) structure (you've glued two smooth manifolds together along a piece); however, one can smooth the attaching map canonically (formally, this is a manifold with corners) and make this a smooth construction.

[edit] Example

As a simple example, consider cutting two disjoint open disks out of the 2-sphere S2, and attaching the cylinder S^1 \times D^1 by gluing each of its ends to one of the boundary circles. The resulting space is topologically the torus S^1 \times S^1.

[edit] Examples

1. Surgery on the Circle

Fig. 1
Fig. 1

As per the above definition, a surgery on the circle consists of cutting out a copy of S^0 \times D^1 and glueing in S^0 \times D^1. The pictures in Fig. 1 show that the result of doing this is either (i) S1 again, or (ii) two copies of S1.

Fig. 2a
Fig. 2a
Fig. 2b
Fig. 2b

2. Surgery on the 2-Sphere

In this case there are more possibilities, since we can start by cutting out either S^0 \times D^2 or S^1 \times D^1.

  • (a) S^1 \times D^1: If we remove a cylinder from the 2-sphere, we are left with two disks. We have to glue back in S^0 \times D^2 - that is, two disks - and it's clear that the result of doing so is to give us two disjoint spheres. (Fig. 2a)
Fig. 2c. This shape cannot be embedded in 3-space.
Fig. 2c. This shape cannot be embedded in 3-space.
  • (b) S^0 \times D^2: Having cut out two disks S^0 \times D^2, we glue back in the cylinder S^1 \times D^1. Interestingly, there are two possible outcomes, depending on whether our glueing maps have the same or opposite orientation on the two boundary circles. If the orientations are the same (Fig. 2b), the resulting manifold is the torus S^1 \times S^1, but if they are different, we obtain the Klein Bottle (Fig. 2c).

[edit] Cobordism and obstructions

There is a close connection between the notion of surgery and that of cobordism. In short, the idea is that given manifolds M and N of equal dimension, there is an explicit correspondence between surgeries from M to N, on one hand, and cobordisms (W;M,N) between the two manifolds, on the other, since every cobordism is the union of the traces of surgeries. This allows for a fruitful exchange of techniques and results between surgery theory and Morse theory.

For example, the h-cobordism theorem states that, in sufficiently high dimension, any h-cobordism between two simply-connected manifolds M and N is trivial (i.e., diffeomorphic to M \times [0,1]), and hence M and N are diffeomorphic. One method proof is obtained via surgery.

More generally, one can attempt via surgery theory to find suitable algebraic and topological obstructions to the existence of a homeomorphism, diffeomorphism, or cobordism between two homotopy-equivalent manifolds (in which term are included topological manifolds). For instance, the s-cobordism theorem states that, in sufficiently high dimension, the only obstruction to a h-cobordism (W;M,N) being trivial is an element of the Whitehead torsion group W(π1(M)).

[edit] Organizing questions

Loosely, the organizing questions of surgery theory are:

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

More formally, one must ask whether up to homotopy:

  • Does a space X have the homotopy type of a smooth manifold?
  • Is a homotopy equivalence f\colon M \to N between two smooth manifolds homotopic to a diffeomorphism?

Note that surgery theory does not give a complete set of invariants to these questions. Instead, it is obstruction-theoretic: it classifies a number of primary obstructions, and the final obstruction is the surgery obstruction.

[edit] Basic structure

What structure beyond homotopy type does a manifold have? At the very least, Poincaré duality.

When is a homotopy equivalence of manifolds homotopic to a homeomorphism? For homotopy spheres this is a generalization of the Poincaré conjecture, proved for dimensions >4 in 1961 by Steve Smale. and in dimension =4 in 1982 by Michael Freedman.

When is a degree 1 normal map from a manifold to a space with Poincaré duality bordant to a homotopy equivalence?

The classification of the exotic spheres by John Milnor and Michel Kervaire in 1963 used surgery on simply-connected manifolds in dimension >5, in which the fundamental group is trivial.. The subsequent Browder-Novikov-Sullivan-Wall surgery theory for manifolds of dimension >4 was developed in the 1960's to answer these questions in terms of a 2-stage obstruction theory, with a primary obstruction in the topological K-theory of vector bundles and a secondary obstruction in the algebraic L-theory of quadratic forms over the ring of the fundamental group π1M. Originally developed for differentiable manifolds the theory was extended to topological manifolds in 1970 by Rob Kirby and Larry Siebenmann. The theory has been applied and extended since 1970, and is now the standard method of classification for manifolds of dimension >4.

[edit] See also

[edit] People

[edit] References

  • Milnor, John: Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow Princeton University Press, Princeton, N.J.

[edit] Introductions

The classic introductory texts are:

  • Browder, Surgery on simply connected manifolds
  • Wall, C.T.C.: Surgery on compact manifolds. 2nd ed., Mathematical Surveys and Monographs 69, A.M.S.

A modern (and easier) introduction, with background, is:

[edit] External links