## Definition

Suppose are differential manifolds, and are smooth maps. Then, a smooth cobordism from to is the following data:

- A smooth cobordism from to i.e. a manifold with boundary with a diffeomorphism
- A smooth map such that the composite of with the inclusions of are respectively the maps

Note that we may sometimes fix the cobordism between and , and ask whether we can *fill in* a function for the particular cobordism.