We know
$$\pi_1(SO(\infty))=\pi_1(RP^3)=\mathbb{Z}_2$$
How to show $$\pi_1(SO(\infty))=\mathbb{Z}_2?$$ and $$\pi_9(SO(\infty))=\mathbb{Z}_2?$$
A related post here.
2026-04-01 20:23:59.1775075039
Show that $\pi_1(SO(\infty))=\mathbb{Z}_2$ and $\pi_9(SO(\infty))=\mathbb{Z}_2$
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Bott periodicity implies that $\pi_k(O(\infty))=\pi_{k+8}(O(\infty))$. From the perspective of homotopy groups, the difference between $O$ and $SO$ is negligible since $SO$ is a connected component of $O$. https://en.wikipedia.org/wiki/Bott_periodicity_theorem