cohomology group of $SO(n)$

I am computing the Alexander-Spanier cohomology $H^i(SO(n),\mathbb{Z})$. I embedded $SO(n)$ into $R^{n^2}$. Since the embedding $i$ is a monomorphism, the induced group homomorphism $i^*$ is an epimorphism. Since $R^{n^2}$ is homotopic to a point, $H^i(R^{n^2})=0 , \forall i \in \mathbb{Z}^+$. That gives us $H^i(SO(n))=0, \forall i \in \mathbb{Z}^+$.

I don't know where I am wrong, can anyone give any suggestions?

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2022-07-25 20:46:55
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