Show that $\pi_1(SO(\infty))=\mathbb{Z}_2$ and $\pi_9(SO(\infty))=\mathbb{Z}_2$

Question

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.

You mean without using Bott periodicity? – Cheerful Parsnip Apr 28 '17 at 01:26 — Cheerful Parsnip, Apr 28 '17 at 01:26
Whatever argument you have is fine – wonderich Apr 28 '17 at 04:21 — wonderich, Apr 28 '17 at 04:21

score 2 · Answer 1 · answered Apr 28 '17 at 17:35

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

Show that $\pi_1(SO(\infty))=\mathbb{Z}_2$ and $\pi_9(SO(\infty))=\mathbb{Z}_2$

1 Answers1