# Relative homology of $D^2$ and $S^1$

I am going crazy showing something that is clearly wrong, but I can't see the error in my logic.

I am calculating $H_1(D^2,S^1)$

Now the disc is contractible so $H_n(D^2)=0$ for all $n$. Also $H_n(S^1)=\mathbb{Z}$ for $n=0,1$ and is $0$ otherwise.

Now we have a long exact sequence of homology groups

$\cdots H_1(D^2)\to H_1(D^2,S^1) \to H_0(S^1) \to H_0(D^2) \cdots$

Using the above results we have the sequence

$0 \to H_1(D^2,S^1) \to \mathbb{Z} \to 0$

and hence $H_1(D^2,S^1) \simeq \mathbb{Z}$

This is clearly not true as this would imply that $H_1(S^2)=\mathbb{Z}$, but I can't identify the gap in my logic (note that I am not working in reduced homology)

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$H_n(D^2)=0$ for all positive $n$...

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thanks...damn 0-th homology groups – Juan S Mar 8 '11 at 23:57