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I'm trying to prove that a reduced homology theory can be define from an unreduced one, but the problem is to define a border map for the reduced homology using the unreduced homology groups.

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Huh? What's a "border map" -- you mean boundary map in homology? $\tilde{H}_n=H_n$ for $n>0$ and $\tilde{H}_0\oplus\mathbb{Z}=H_0$. –  Chris Gerig Jul 15 '12 at 10:05

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You might know, Hatcher uses the maps

$\overline H_n(X)=\ker(H_n(X)\rightarrow H_n(point))$

$H_n(X)=\overline H_n(X\sqcup point)$

It's left as an exercise to check that these two maps are inverses of each other as transformation of the unreduced and the reduced homology theories.

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Hello Aneesh! Hatcher gives maps but I don't know if theese really define homology theories, I want to prove this before the equivalence.. –  mary-j Jul 15 '12 at 13:53

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