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 must be null homotopic
- For a homotopy equivalence 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.
More generally, its generalizations serve as replacements for the (unstable) tangent bundle.
Read more about this topic: Stable Normal Bundle