Smooth partitions of unity and differentiable good open covers are important technical tools in the realm of smooth manifolds.

Are there analogs of these tools for piecewise linear manifolds?

A PL good open cover of a PL-manifold is a locally finite open cover {Ui} such that every finite intersection of Ui is either empty or PL-isomorphic to Rn.

A PL partition of unity subordinate to an open cover {Ui} of a PL-manifold X is a family of nonnegative PL-functions fi: X→R such that supp fi is a subset of Ui, supp fi form a locally finite family, and the sum of fi is 1.

Do PL-manifolds admit PL good open covers? Do open covers of PL-manifolds admit subordinate PL partitions of unity?

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    $\begingroup$ Stupid question - to find a good open cover can I just: fix a triangulation, barycentrically subdivide a few times, and then take the stars of top-dimensional simplices? $\endgroup$
    – Sam Nead
    Oct 28, 2014 at 19:42
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    $\begingroup$ Yes, certainly they have PL partitions of unity. For smooth manifolds all you need for the proof is small bump functions, which have nice piecewise-analytic explicit formulas. On a PL manifold you can manufacture your bump functions much more easily -- take a suitably-refined triangulation, force the function to be 1 on a top simplex, 0 away from a small neighbourhood and linearly interpolate the rest. The remainder of the proof is just like the smooth category. $\endgroup$ Oct 28, 2014 at 19:46
  • $\begingroup$ It seems to me like PL partitions of unity would be less useful though, as you rarely have useful vector bundles to work with. $\endgroup$ Oct 28, 2014 at 19:50
  • $\begingroup$ @SamNead: Is it obvious that all finite nonempty intersections are isomorphic to R^n in your construction? $\endgroup$ Oct 28, 2014 at 23:15
  • $\begingroup$ I think that all intersections are again open stars (union of open simplices whose closures contain a given simplex). Thus the axioms of a PL manifold should take care of me. (A star is the open cone on a link, and all links are PL spheres of the correct dimension.) I'm by no means an expert... $\endgroup$
    – Sam Nead
    Oct 29, 2014 at 17:42


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