We consider as the group of all rotations about the origin of under the operation of composition. Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation.
We consider as the unit 3-sphere with antipodal points identified.
Consider the map from the unit ball in to , which maps to the rotation about through the angle (and maps to the identity rotation). This mapping is clearly smooth and surjective. Its restriction to the interior of is injective since on the interior . On the boundary of , two rotations through and through are the same. Hence the mapping induces a smooth bijective map from , with antipodal points on the boundary identified, to . The inverse of this map, is also smooth. (To see that the inverse is smooth, write . Then , and so exists and is continuous for all orders . Similar results hold for the variables and , and also mixed partials. By multivariable chain rule, one can see that all component functions are indeed smooth, so the inverse is smooth as claimed.)
Hence , the unit ball in with antipodal points on the boundary identified.
Next, the mapping is a diffeomorphism between and the upper unit hemisphere of with antipodal points on the equator identified. The latter space is clearly diffeomorphic to . Hence, we have shown