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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 )??

Please help me.

Thank you

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1 Answer 1

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

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