**manifolds** and **cobordisms**

cobordism theory, *Introduction*

Given two manifolds $X$, $Y$ (e.g. topological manifolds, differentiable manifolds, smooth manifolds, etc.) the *product manifold* $X \times Y$ is the Cartesian product in the corresponding category of manifolds: its underlying topological space is the product topological space and its charts are the Cartesian product of the given charts of $X$ and $Y$.

For Cartesian spaces we have

$\mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \simeq \mathbb{R}^{n_1 + n_2}$

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