Stable normal bundle

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In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. It is also called the Spivak normal bundle, after Michael Spivak (reference below). There are analogs for generalizations of manifold, notably PL-manifolds, topological manifolds, and Poincaré spaces.

1 Construction via embeddings
2 Construction via classifying spaces
3 Why normal?
4 Applications
5 References

[edit] Construction via embeddings

Given an embedding of a manifold in Euclidean space, it has a normal bundle. The embedding is not unique, but for high dimension it is unique up to homotopy, thus the (class of) the bundle is unique, and called the stable normal bundle.

This construction works for any Poincaré space X: a finite CW-complex admits a stably unique (up to homotopy) embedding in Euclidean space, via general position, and this embedding yields a spherical fibration over X. For more restricted spaces (notably PL-manifolds and topological manifolds), one gets stronger data.

[edit] Construction via classifying spaces

An n-manifold M has a tangent bundle, which has a classifying map (up to homotopy)

$\xi\colon M \to BO(n)$

Composing with the inclusion $BO(n) \to BO$ yields (the homotopy class of a classifying map of) the stable tangent bundle; taking the dual yields the stable normal bundle. (Or equivalently, dualizing and then stabilizing.)

[edit] Why normal?

Stable normal data is used instead of unstable tangential data because generalizations of manifolds have natural stable normal-type structures, but not unstable tangential ones.

A Poincaré space X does not have a tangent bundle, but it does have a well-defined stable spherical fibration, which for a differentiable manifold is the spherical fibration associated to the stable normal bundle; thus a primary obstruction to X having the homotopy type of a differentiable manifold is that the spherical fibration lifts to a vector bundle.

In classifying space language, the stable spherical fibration $X \to BH$ must lift to $X \to BG$ , which is equivalent to the map $X \to B(G/H)$ being null homotopic; recall the distinguished triangle:

$BG \to BH \to B(G/H)$

Thus the bundle obstruction to the existence of a (smooth) manifold structure is the class $X \to B(G/H)$ .

[edit] Applications

The stable normal bundle is fundamental in surgery theory as a primary obstruction:

For a Poincaré space X to have the homotopy type of a smooth manifold, the map $X \to B(G/H)$ must be null homotopic
For a homotopy equivalence $f\colon M \to N$ between two manifolds to be homotopic to a diffeomorphism, it must pull back the stable normal bundle on N to the stable normal bundle on M