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It is a well-known fact that for every compact oriented odd-dimensional manifold $\mathcal{M}$ with boundary it holds that $$\chi(\mathcal{M})=\frac{1}{2}\chi(\partial\mathcal{M}).$$ In particular, if ...

Edit: According to the interesting comments of Michael Albanese and Nick L we revise the question as follows: By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ ...