Recall that elements of all take the form with . Therefore is topologically , the 3-dimensional hypersphere. (This is enough to show that is simply connected.) Because the kernel is (verify!), antipodal points are identified on mapping into , so is homeomorphic to real projective 3-space.
This can be seen another way. A rotation can be specified by a vector along the axis of the rotation, with magnitude giving the angle of the rotation. This serves to identify elements of with points inside or on a ball of radius . However, antipodal points on the surface of the ball represent the same rotation. The resulting space (a three-dimensional ball with antipodal points on the surface identified) is well-known to be homeomorphic to real projective 3-space. (If you think about this argument for a bit, you should see an implicit use of the exponential mapping from into .)
A loop in the topological space can be visualized as a continous ``trajectory'' of rotations: we take a rigid object and turn it around in some fashion, finally restoring it to its original orientation. The following fact can be deduced from this: if a solid object is connected by threads to a surrounding room, and the object is turned through , then the threads can be untangled without turning the object any more. However, if the object is turned through , then the threads cannot be untangled. (The two-thread version of this is known as ``Dirac's string trick''.) In this sense, a continuous turn through is not the same as no turn at all (but a turn is.)
© 2001 Michael Weiss