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De Rham's first theorem

  1. Aug 17, 2011 #1

    ivl

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    Hi all!
    I have a possibly trivial (possibly non-trivial? :rofl:) question. Here it is:

    Assumption-Assume I have a closed p-form, whose integral over any p-cycle is always zero.
    Statement-The closed p-form is also exact, by what is sometimes called de Rham's first theorem

    My question is: what are the topological implications of my statement? (e.g. am I implying that all p-cycles are p-boundaries??)

    Further question: de Rham's theorem is often proved for compact manifolds. Is my statement true even for manifolds which are not compact? (assume my manifold is paracompact, but not compact).

    Thanks a lot!
     
  2. jcsd
  3. Aug 17, 2011 #2

    mathwonk

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    I am not an expert but my reading gives me the following impressions:

    The first statement seems to be true for all 2nd countable hausdorff manifolds, i.e. all smooth manifolds normally considered by the average person.

    the proof is given in the book of singer and thorpe, under the assumption there is a smooth triangulation, and with simplicial cohomology. any manifold that has a closed embedding in euclidean space, i.e. any manifold as above, has such a triangulation according to whitney.

    it also has partitions of unity which implies it is paracompact.

    bott tu give the proof assuming a "good cover" of the manifold, which also follows from triangulability.

    good references include bott -tu, singer and thorpe, morris hirsch, a. weil (comm. math. helvetici about 1952), spivak differential geometry vol. 1, chap. 11, problem 14, and the book of frank warner for an especially complete version although rather abstract in terms of shaves.


    the answer to:

    "My question is: what are the topological implications of my statement? (e.g. am I implying that all p-cycles are p-boundaries??)"

    is no. since the earlier statement is true for all manifolds it implies nothing at all about the topology of the manifold. however if you were to change your earlier statement to say that all closed p forms are exact, it would then imply your second statement. i.e. by de rham, all closed p forms are exact if and only if all p cycles are boundaries.

    another reference of course is the book of georges de rham, but i have not read it.
     
  4. Aug 17, 2011 #3

    ivl

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    Excellent answer, thanks!
     
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