Results 1 to 3 of 3

Thread: Homology & path components

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    87

    Homology & path components

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

    What I know:

    - $\displaystyle H_0(X)$ 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!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by harbottle View Post
    I have to show that $\displaystyle H_0(X,A) = 0$ iff A meets each of the path-components of X.

    What I know:

    - $\displaystyle H_0(X)$ 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

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

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

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

    Since $\displaystyle H_0(A) \rightarrow H_0(X)$ is surjective, it forces N is a zero map and $\displaystyle H_0(X,A)=0$ (The kernel of N should be the whole image of M, which is H_0(X)) .
    Last edited by aliceinwonderland; Oct 24th 2009 at 01:54 PM. Reason: cor.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    Posts
    87
    Thank you very much! That makes sense.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Homology Groups
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Dec 13th 2010, 01:33 PM
  2. Homology group of S
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: May 28th 2010, 05:11 AM
  3. Homology of the 3-Torus
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 30th 2009, 02:32 AM
  4. Path connected components.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 22nd 2009, 04:46 AM
  5. degree of a path, path homotopy.
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Mar 19th 2009, 05:37 PM

Search Tags


/mathhelpforum @mathhelpforum