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Math Help - Homology and Cohomology

  1. #1
    Senior Member Sampras's Avatar
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    Homology and Cohomology

    What exactly is the difference between homology and cohomology? We know that homology is the number of  n -cycles that are not  n -boundaries. This is, essentially the number of holes in a simplicial complex  X . What is the geometrical interpretation of cohomology?
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  2. #2
    Senior Member
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    Cohomology classes are exact forms which are not closed. If you integrate a function along a closed surface, the integral depends only on the cohomology class of the integrand. Loosely, I tend to hear that cohomology measures obstructions to creating global solutions from local solutions, say of an equation dx=y. The cohomology class of y measures how this equation fails to have a global solution.

    I think that cohomology is less geometric than homology, but the ring structure makes it more useful in many situations.
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