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?