Cobordism of smooth maps

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Definition

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:

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.