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Math Help - calculation of cohomology group for a 0-dim manifold

  1. #1
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    calculation of cohomology group for a 0-dim manifold

    Could someone explain to me this example? I know all the basis and definitions but I am new to calculation of cohomology group

    here is the example

    http://i51.tinypic.com/wb7609.jpg

    Why  Z^{k}= \begin{cases} 0 for \mathbb{R} , k=0\\ 0 , k>0\end{cases}?? why do we get a real numbers and then we get just 0?

    and then boundary group  B^{k}(M)=0 I guess that it is because a point does not have a boundary, is it correct and finally how did they get the cohomology group? how did they obtain again R and 0?

    Thank you
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rayman View Post
    Could someone explain to me this example? I know all the basis and definitions but I am new to calculation of cohomology group

    here is the example

    http://i51.tinypic.com/wb7609.jpg

    Why  Z^{k}= \begin{cases} 0 for \mathbb{R} , k=0\\ 0 , k>0\end{cases}?? why do we get a real numbers and then we get just 0?

    and then boundary group  B^{k}(M)=0 I guess that it is because a point does not have a boundary, is it correct and finally how did they get the cohomology group? how did they obtain again R and 0?

    Thank you
    What are your Z^k, I can't seem to find their definition on their. Are they your boundary maps? Or, perhaps the kernels of your boundary maps?
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  3. #3
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     Z^{k} is just a cyclic group or in the language of cohomology  ker_{d+1}
    boundrary is the other one
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rayman View Post
     Z^{k} is just a cyclic group or in the language of cohomology  ker_{d+1}
    boundrary is the other one
    So, Z^k is the free abelian group with rank k?
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    yes
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rayman View Post
    yes
    But then your question doesn't make sense? How could \mathbb{Z}^0\cong\mathbb{R}?
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    well that is an example taken from the text book and it seems it can be, I will investigate this and if I come to some good conclusions I will write again
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  8. #8
    Senior Member Tinyboss's Avatar
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    Are you sure it's not the 0-th cocycle group?
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  9. #9
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    yes, you are right this is 0-th cocycle group
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