Theorem 22.1

The unit sphere $S^3$ is the double cover of the space $SO(3, \mathbb{R})$ and as a topological space is homeomorphic to $\mathbb{R}P^3$. In particular $\pi_1(SO(3, \mathbb{R}))$ is the cyclic group of order two.

The proof will be split into several lemmas. We begin by setting up a few notations which would remain in force throughout the lecture. We shall regard $S^3$ as the set of all unit quaternions forming a subgroup of the multiplicative group of non-zero quaternions.