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If I start with a, say, 3-CW complex $X$ which can be embedded in $\mathbb{R}^5$, I can get a neighbourhood $U$ of $X$ which has the same homotopy type of $X$. Then $U$ is a $5-$ dimensional open ...

Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex. Does $M$ admit a finite good cover? (i.e. a finite cover by contractible ...