Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Definition of bordism - gluing manifolds with structure

In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U ...