Can you find mistake in my computation of $H^{k}(S^{n})$.
Sphere is disjoint union of two spaces: $S^{n} = \mathbb{R}^{n}\sqcup\mathbb{R^{0}},$ so $H^{k}(S^n) = H^{k}(\mathbb{R}^{n})\oplus H^{k}(\mathbb{R^{0}}).$ In particular $H^{0}(S^{n}) = \mathbb{R}\oplus\mathbb{R}=\mathbb{R}^{2}$ and $H^{k}(S^{n}) = 0,~~~k>0.$
Where is mistake? Thanks a lot!