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. More technically, the rough 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 and homotopy (and/or other interesting invariants) of the manifold are known.

1 Definition
2 Examples
3 Cobordism and obstructions
4 See also
5 References
6 External links

[edit] Definition

The intuitive idea of surgery is simple. Starting with a given manifold $M$ of dimnesion $n$ , we cut out some embedded $n$ -dimensional pieces, then attach an $n$ -dimensional 'handle' along the resultant boundary. As a simple example, consider cutting two disjoint open disks out of the 2-sphere $S 2$ , and attaching the cylinder $S^1 \times D^1$ by glueing each of its ends to one of the boundary circles. The resulting space is topologically the torus $S^1 \times S^1$ .

However, the problem with this intuitive approach is that the space we end up with doesn't necessarily have a smooth structure - that is, it's not a manifold. So, to perform surgery 'smoothly', we need a more technical definition.

(Definition to be added.)

[edit] Examples

1. Surgery on the Circle

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 (to be added) show that the result of doing this is either (i) $S 1$ again, or (ii) two copies of $S 1$ .

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.

(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, the resulting manifold is the torus $S^1 \times S^1$ , but if they are different, we obtain the Klein Bottle.

(To be expanded)

[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 correnspondence between surgeries from $M$ to $N$ , on one hand, and cobordisms $(W; M, N)$ between the two manifolds, on the other. 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 $W (π 1 (M))$ .

[edit] See also

[edit] References

Milnor, John: Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow Princeton University Press, Princeton, N.J.
Ranicki, Andrew: Algebraic and Geometric Surgery. Oxford Mathematical Monograph, OUP.
Wall, C.T.C.: Surgery on compact manifolds. 2nd ed., Mathematical Surveys and Monographs 69, A.M.S.