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nLab > nLab General Discussions: Good open covers and partitions of unity for PL-manifolds

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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 28th 2014

    Is anything known about the existence of partitions of unity and good open covers for PL-manifolds?

    Here a good open cover of a PL-manifold is a locally finite open cover {U_i} such that every finite intersection of U_i is either empty or PL-isomorphic to R^n.

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

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeOct 28th 2014
    • (edited Oct 28th 2014)

    I do not know if it helps but there is a MO question which recalls that PL-manifolds are combinatorial manifolds therefore simplicial complexes. Any open cover of such can be refined to give the nerve is the same as a subdivision of the underlying simplicial complex. That does not give you partitions of unity, but may be of use.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 29th 2014

    I asked on MathOverflow (http://mathoverflow.net/questions/185623/are-there-analogs-of-smooth-partitions-of-unity-and-good-open-covers-for-pl-mani) and it seems like partitions of unity are easy to construct, and there is a candidate construction for good open covers, though in this case it’s less clear that it works.

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