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Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\alpha_i}$ is contractible; or (2) it is locally-finite

Then, can we find a triangulation $\{\Delta_i\}$ of $M$ satisfying the following: for each $i$ there exists a $U_\alpha$ such that whenever $\Delta_j\cap \Delta_i \neq \varnothing$, we have $\Delta_j\subset U_\alpha$?

If this is not true, can we add some further conditions to make it true? If this is true, can you give some hints of proof or some reference? (I guess this should be true at least for compact $M$) Thanks!

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  • $\begingroup$ arxiv.org/abs/1311.0117 $\endgroup$
    – Vidit Nanda
    Commented Oct 18, 2018 at 21:27
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    $\begingroup$ For compact manifolds isn’t this just a simple Lebesgue number argument? Fix a metric on the manifold and let e be the Lebesgue number of the cover. Barecentrically subdividing a triangulation enough, we can assume that each simplex has diameter at most e/3. This implies that for all simplices, the e-ball aound a point in the simplex contains it and all simplices touching it. But by the definition of the Lebesgue number the e-ball is contained in some set in the cover. $\endgroup$
    – Andy Putman
    Commented Oct 19, 2018 at 3:48

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