Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Conceptual Question on de Rham cohomology.

  1. Jan 1, 2012 #1
    Hi everybody,

    Currently, I am studying cohomology on my own. I have a question:

    Why H rD(M) = 0, when r > n

    n is the dimension of manifold M
    My book says it is obvious, but to me it is not obvious.

    I wish someone could explain this question to me.
     
  2. jcsd
  3. Jan 1, 2012 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Well, how is the group defined?
     
  4. Jan 1, 2012 #3
    The group is defined as
    HrD (M) = Ker(dr)/Im(dr-1)
     
  5. Jan 1, 2012 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    And what groups is dr a homomorphism from and to?
     
  6. Jan 2, 2012 #5

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    it follows from properties of the wedge product, as is being suggested.
     
  7. Jan 12, 2012 #6

    Bacle2

    User Avatar
    Science Advisor

    How do you define n-cocycles and n-coboundaries?
     
  8. Jan 12, 2012 #7

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    there aren't even any ≠0 cochains in dimensions above the dimension of the manifold.

    the reason is essentially that an nbyn determinant is always zero if the matrix has rank < n.
     
  9. Jan 12, 2012 #8

    Bacle2

    User Avatar
    Science Advisor

    Yes, that was the point I was trying to make. Look up the definition of n-cocycles and n-coboundaries to see what the cohomology groups are . Or, if you have the right conditions for Poincare Duality, see why you cannot have (n+k)-cycles; k>0, in an n-manifold.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: A Conceptual Question on de Rham cohomology.
  1. De Rham cohomology (Replies: 10)

Loading...