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What are the homotopy groups of $O(d,d)$? Is it possible to compute them somwhow by using the fact that $O(d)\times O(d)$ is the maximal compact subgroup, and the homotopy groups of $O(d)$ are known?

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If $G$ is a Lie group and $K$ is a maximal compact subgroup then the inclusion $K \to G$ is a homotopy equivalence, and hence they have the same homotopy groups. Therefore

$$ \pi_kO(d, d) \cong \pi_k (O(d)\times O(d)) \cong \pi_kO(d) \times \pi_kO(d) $$

Then $\pi_k O(d)$ is known in the stable range by Bott Periodicity, and in the meta-stable range there is work done by Kervaire and Lundell and others, but I think for $k >> d$ these groups are not known in general.

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