# Tangential homotopy equivalences (Ex)

From Manifold Atlas

A homotopy equivalence of closed smooth -manifolds is called *stably tangential* if ; i.e. pulls back the stable tangent bundle of to .

**Example 0.1.**

- Give an example of a stably tangential homotopy equivalence which is not homotopic to a diffeomorphism.
- Show that if is a stably tangential homotopy equivelance, then there is a diffeomorphism

such that is homotopic to , where is the projection.