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Math Help - Homology & path components

  1. #1
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    Homology & path components

    I have to show that H_0(X,A) = 0 iff A meets each of the path-components of X.

    What I know:

    - H_0(X) is the direct sum of n copies of Z where n is the # of path-components in X
    - A will have at least as many path-components as X

    What I don't know:

    - How to use relative Homology to prove the result

    Help!
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  2. #2
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    Quote Originally Posted by harbottle View Post
    I have to show that H_0(X,A) = 0 iff A meets each of the path-components of X.

    What I know:

    - H_0(X) is the direct sum of n copies of Z where n is the # of path-components in X
    - A will have at least as many path-components as X

    What I don't know:

    - How to use relative Homology to prove the result

    Help!
    The long exact sequence has the following form

    ...H_0(A) \rightarrow H_0(X) \rightarrow H_0(X,A) \rightarrow 0 (Hatcher p 115).

    Since A has at least as many path-components as X, H_0(A) \rightarrow H_0(X) is surjective (They are both free abelian groups and H_0(A) has the same or higher rank than H_0(X)).

    Let M:H_0(A) \rightarrow H_0(X) and N:H_0(X) \rightarrow H_0(X, A). By the definition of the long exact sequence, \text{Im M = Ker N}.

    Since H_0(A) \rightarrow H_0(X) is surjective, it forces N is a zero map and H_0(X,A)=0 (The kernel of N should be the whole image of M, which is H_0(X)) .
    Last edited by aliceinwonderland; October 24th 2009 at 02:54 PM. Reason: cor.
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  3. #3
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    Thank you very much! That makes sense.
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