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Definition of a 3-cobordism (in my context) is a pair $(M, \partial_{-} M, \partial_{+}M)$, where $M$ is a closed orientable topological 3-manifold and $\partial M$ is a disjoint union of $\partial_{-} M$ and $\partial_{+}M)$.

I have a question regarding the following sentence:

"$(W, U, V)$ is a 4-dimensional cobordism with boundary $(M, \partial_{-} M, \partial_{+}M)$."

Here $W$ is a 4-dimensional manifold and $U, V$ are 3 dimensional manifolds.

  1. What is the definition of 4-dimensional cobordism used here? Is it just same as the 3-dimensional case? That is, $\partial W$ is a disjoint union of $U$ and $V$.

  2. Also what is the definition of the term boundary used above?

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