Stable Normal Bundle - Motivation

Motivation

There is no intrinsic notion of a normal vector to a manifold, unlike tangent or cotangent vectors – for instance, the normal space depends on which dimension one is embedding into – so the stable normal bundle instead provides a notion of a stable normal space: a normal space (and normal vectors) up to trivial summands.

Why stable normal, instead of stable tangent? Stable normal data is used instead of unstable tangential data because generalizations of manifolds have natural stable normal-type structures, coming from tubular neighborhoods and generalizations, but not unstable tangential ones, as the local structure is not smooth.

Spherical fibrations over a space X are classified by the homotopy classes of maps to a classifying space, with homotopy groups the stable homotopy groups of spheres

The forgetful map extends to a fibration sequence

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, i.e. the Spivak spherical fibration must lift to, which is equivalent to the map being null homotopic Thus the bundle obstruction to the existence of a (smooth) manifold structure is the class . The secondary obstruction is the Wall surgery obstruction.