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Does anyone know a good trick to computing homology groups of the sphere minus the wedge of two spheres of possibly different dimension $S^n \setminus S^k\vee S^\ell$ ? Any particular $k$ and $\ell$ is not so bad, but the general case has so many cases. Can one avoid this with a some sneaky exact sequence?

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As Grumpy Parsnip suggests we can take Alexander duality so that

\begin{align*} \tilde H_i(S^n - S^k \vee S^\ell; \mathbb Z) &\cong \tilde H^{n-i-1}(S^k \vee S^\ell; \mathbb Z)\end{align*} which is $\mathbb Z$ for $i=n-1-k$ or $n-1-\ell$ and trivial otherwise.

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