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Thread: About a short exact sequence of relative homology groups.

  1. May 24th 2011, 05:53 AM #1
    Capillarian
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    About a short exact sequence of relative homology groups.

    Let x be a point in X. My question is about the following short exact sequence:

    0 \to H_1(X) \to H_1(X,\{x\}) \to H_0(\{x\}) \: \xrightarrow{i_\ast} \: H_0(X)\to H(X,\{x\})\to 0,

    where i_* : H_0 (\{ x \} ) \to H_0(X) is the homomorphism induced by the obvious inclusion. I am told that i_* is injective. Although this does make some sense intuitively, I can't really prove it. Can someone show me why it is?
    Last edited by Capillarian; May 24th 2011 at 09:18 AM.
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