# Showing that there is not a global diffeomorphism between unit quaternions and $\mathrm{SO}(3)$

I am new to differential geometry. I have the following question:

Let $$\mathbf{Q}$$ denote the set of unit quaternions. I already have proved using the implicit function theorem that $$\mathbf{Q}$$ is a $$3$$-manifold. Now, I need to show:

There does not exist a global diffeomorphism $$\phi$$ between $$\mathbf{Q}$$ and $$\mathrm{SO}(3)$$, i.e., show that there does not exist $$\phi: \mathbf{Q} \to \mathrm{SO}(3)$$, such that $$\phi$$ is a $$C^\infty$$ bijection.

Any ideas of how to go about it?

• Here's an idea: the set of unit quaternions is $S^3$. But you can see that $\mathrm{SO}(3)$ is diffeomorphic to $\mathbb{R}P^3$. Now try to find something that distinguishes $\mathbb{R}P^3$ from $S^3$. They have different fundamental groups, but I don't know if that helps in manifolds. – stressed out Feb 17 '19 at 23:36
• As a side note, I just checked on the internet that the De Rham cohomologies of $\mathbb{R}P^3$ and $S^3$ are the same unfortunately. – stressed out Feb 17 '19 at 23:53
• As a note, global diffeomorphism is not the same as a $C^\infty$ bijection, you also require its inverse to be $C^\infty.$ – positrón0802 Feb 18 '19 at 0:11

The set of unit quaternions is $$S^3$$. It can be shown using the algebra of quaternions that $$q: S^3 \to \mathrm{SO}(3)$$ is a double cover with the kernel $$\{\pm 1\}$$. This means that $$\mathrm{SO}(3)$$ is diffeomorphic to $$S^3/\{\pm 1\}$$ which is $$\mathbb{R}P^3$$. But the fundamental group of $$n$$-sphere is trivial for $$n>1$$ while the fundamental group of $$\mathbb{R}P^n$$ for $$n>1$$ is $$\mathbb{Z}_2$$. This means that they can't be diffeomorphic because homeomorphisms preserve fundamental groups.
Topologically the set of unit quaternions is nothing else than the ordinary $$3$$-sphere $$S^3 \subset \mathbb{R}^4$$. It is well-known that $$S^3$$ is simply connected wheras $$SO(3)$$ is not simply connected. Hence there cannot exist a homeomorphism (let alone a diffeomorphism) between $$S^3$$ and $$SO(3)$$.