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Ref. Please: Homology Functor on Prod. Spaces ; Chain Groups, Differentials on Product spaces

  1. Oct 6, 2014 #1

    WWGD

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    Hi all,
    Went to a seminar today, arrived a few minutes late; hope someone can tell me something about this topic and/or give a ref so that I can read on it . I know this is a lot of material; if you can refer me to at least some if, I would appreciate it :

    1)Basically, understanding how/why the (co)homology functor can/does fail to agree with the product of spaces, when we do not work with coefficients in a field i.e.,

    ## H^k(X \times Y) \neq \bigoplus_{i+j=k} H^i (X) \otimes H^j(X) ##

    is not satisfied when we do not work with coefficients in a field; I am tooignorant at this point to understand why/how field coefficients matter ( I was considering the issue of torsion, but there was no mention that the fields had to be of characteristic zero). Also,

    2)What kind of correction terms do we need when we consider (co)chain groups on product spaces, i.e., what is the relationship between## C_p(X \times Y)## and ##C_p(X), C_p(Y) ## and how the differential/boundary operator works on the (co)homology of products, i.e., if we have ## \delta ## boundary operator on different (co)chain complexes, how/when can we define a boundary operator on the product ## X \times X ##. Does it make sense to consider a product of (co)chain complexes with different boundary operators?

    Basically, we work with inclusion , diagonal and projection diagrams/operations, i.e., ## x \rightarrow (x,x)##, etc.

    Thanks.
     
    Last edited: Oct 6, 2014
  2. jcsd
  3. Oct 6, 2014 #2

    WWGD

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    I guess I was being lazy; in case anyone is interested, the main topics are algebras, coalgebras, and their dualizations, the existence of coproducts in homology/cohomology, Hopf algebras , etc. Some of these issues and other related ones are dealt, e.g., in : http://mathoverflow.net/questions/415/does-homology-have-a-coproduct

    Still, I would appreciate additional comments/refs.
     
  4. Oct 8, 2014 #3
    Have you read what Hatcher has to say about the chain complexes and homology/cohomology of products (especially for CW-complexes)?

    http://www.math.cornell.edu/~hatcher/AT/ATpage.html

    I tend to like to figure stuff out in the CW setting because it's more geometrical and when all is said and done, everything is weak-equivalent to a CW complex, so you don't lose anything as far as homology and cohomology goes.

    I'm also being a bit lazy, but if I wanted to understand it more fully, I'd go back and read that. It's hard to see what you were missing because I don't know how you convinced yourself that the homology is that way if you have field coefficients. To me, I just know the chain complex works that way and it's not clear that when you take homology that it's going to break down the same way in terms of just the direct sum of tensor products thing that you have there. That only happens on the chain level.
     
  5. Oct 8, 2014 #4

    WWGD

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    There was also the issue of the conditions , if any, under which the product in cohomology , i.e., the cup product can be dualized into a coproduct in homology, and general conditions under which homology and cohomology are duals of each other. It seems like finite-dimensionality and bgeing torsion-free are necessary, thou am not sure they are sufficient. But , yes, I do need to read some more; thanks for the link.
     
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