# Relative homology of ball and sphere

What is the result of $H_k(B^n,S^{n-1}; \mathbb{A })$ and in any book can i found the proof ?

And what about $H_n(S^{n};\mathbb{A})$ (sigular homology of the sphere )??

Thank you

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You have the exact sequence $$\underset{=0}{\underbrace{\tilde{H}_k(B^n)}} \to \tilde{H}_k(B^n,S^{n-1}) \to \tilde{H}_{k-1}(S^{n-1}) \to \underset{=0}{\underbrace{\tilde{H}_{k-1}(B^n)}},$$ so $$\tilde{H}_k(B^n,S^{n-1}) \simeq \tilde{H}_{k-1}(S^{n-1})= \left\{ \begin{array}{cl} \mathbb{A} & \text{if} \ k=n \\ 0 & \text{otherwise} \end{array} \right..$$

If you do not know $\tilde{H}_k(S^n)$, you can use Mayer-Vietoris sequence in order to show $$\tilde{H}_k(S^n) \simeq \tilde{H}_{k-1}(S^{n-1})$$ by decomposing $S^n$ in two hemispheres.

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$\tilde {H}$ is the reduced homology ?, i dont want to use the reduced homology ! – Vrouvrou Feb 21 '14 at 8:55
@Vrouvrou: Why not? In any case that only makes a difference in low degrees, write the exact sequence explicitly and you'll find the same result. – Najib Idrissi Feb 21 '14 at 9:22

A little bit late... (1 year later).

You can also use @Seirios idea inductively, i.e.

$$H_k(B^n,S^{n-1})\overset{\sim}{\longrightarrow}H_{k-1}(S^{n-1})$$

by useing $H_k(B^n,S^{n-1})\simeq \overset{\sim}{H}_k(B^n/S^{n-1},pt.)\simeq \overset{\sim}{H}_k(S^{n})$ (because $B^n/S^{n-1}\cong S^{n}$).

I think the basic idea can be found in Hatcher's book. But I can not remind which page...

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