# Cobordism of smooth maps

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Suppose $M_1, M_2, P$ are differential manifolds, and $f_1:M_1 \to P, f_2:M_2 \to P$ are smooth maps. Then, a smooth cobordism from $f_1$ to $f_2$ is the following data:
• A smooth cobordism from $M_1$ to $M_2$ i.e. a manifold with boundary $(N, \partial N)$ with a diffeomorphism $M_1 \sqcup M_2 \to \partial N$
• A smooth map $F:N \to P$ such that the composite of $F$ with the inclusions of $M_1, M_2$ are respectively the maps $f_1, f_2$
Note that we may sometimes fix the cobordism between $M_1$ and $M_2$, and ask whether we can fill in a function $F$ for the particular cobordism.