# Proving that unit quaternions are a 3 Manifold

I am very new to topology, and I am having trouble on how to prove if something is a manifold or not. The question states that:

Let Q donate the set of unit quaternions (a) Show that Q is a 3-manifold.

In general, I would also like to know how to go about proving if some set is a manifold or not. Thanks!

• What's your definition of manifold? – José Carlos Santos Feb 17 at 22:03
• First, figure out how to define the unit sphere in $\mathbb{R}^n$ as a manifold. I like using stereographic projection to do this. Next, show that the set of unit quaternions is a unit sphere in $\mathbb{R}^4$. – Deane Feb 17 at 22:05
• We are taught the following definition of manifold: A set S is a k-dimensional manifold if it is locally homeomorphic to Rk, meaning that each point in S possesses a neighborhood that is homeomorphic to an open set in Rk – math_seeker Feb 17 at 22:05
• Have you learned the Regular Value Theorem? That's the easy way to show that something like this is a manifold – Nathaniel Mayer Feb 17 at 22:08
• Or maybe you've learned something like it under a different name, like the Implicit Function Theorem? – Nathaniel Mayer Feb 17 at 22:09