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.

1 Definition
2 Example
- 2.1 Examples
- 2.2 Cobordism and obstructions
3 Organizing questions
4 Basic structure
5 See also
6 References
7 External links

[edit] Definition

The intuitive idea of surgery is simple: it is the manifold analog of attaching cells. Specifically, a k-handle is the manifold analog of a k-cell.

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 CW complexes, but you cannot attach a single k-cell to an n-manifold and have it stay a manifold, because you've glued a k-dimension object onto an n-dimensional one. Instead, you must "thicken" the cell up to being n-dimensional. Also, handles are analogous to regular CW-complexes, in that the attaching maps are homeomorphisms.

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 that represents the class $α$ . The analog in surgery is to attach a p-handle.

Given an embedded sphere $S^p \subset M$ , with trivial normal bundle, a tubular neighborhood of it is diffeomorphic to $S^p \times D^q$ where p+q = n. $S^p \times D^q$ is a manifold with boundary, whose boundary is $S^p \times S^{q-1}$ , which is also the boundary of $D^{p+1} \times S^{q-1}$ . Attaching a p-handle along $S^p \subset M$ is defined as cutting out the interior of $S^p \times D^q$ ("hollowing out a bagel"), and then attaching $D^{p+1} \times S^{q-1}$ along the resulting boundary.

Points to note:

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.)
p-surgery fills in a p-hole and opens a (q-1)-hole: thus if p<q-1 (so 2p < n-1; below the middle dimension (rounding down if n is odd)), you can close a p-hole, but

These two points are the technical issues in performing surgery; the latter foreshadows that the action in surgery is in the middle dimension.

As stated, the surgered manifold has a PDIFF (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 $S 2$ , 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

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] 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?

What structure beyond homotopy class does a homotopy equivalence of manifolds have?

Working down from a manifold, or up from a topological space

Given a manifold $M$ ,

fundamental group $π 1 M$
orientation character $\pi_1 M \to \left\{\pm 1\right\}$
stable normal bundle

Thus for a space to have the homotopy type of a smooth manifold, Poincaré space

A homotopy equivalence of manifolds:

Whitehead torsion (simple homotopy type)
Fundamental class
normal data

Given a simple degree 1 normal map...

[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.