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I am looking for a simple proof of a Mostow Theroem, which asserts that any connected Lie group $G$ admits a maximal compact subgroup $K$ (which is necessarily connected) such that $$G\simeq K\times\mathbb{R}^d\quad(\text{for certain}\ d).$$ So $G$ and $K$ are homotopy equivalent, so they have exactly the same cohomology groups.

I wonder if there is a 'simple' proof of the above theorem.

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Have you seen…? – Akhil Mathew Sep 2 '11 at 12:47
Thank you Akhil, for the nice link. It's not trivial, but I try to study it in details! – amine Sep 4 '11 at 10:40

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