Here is a video animation that illustrates the topology of the rotation group SO(3), which is the group of rotations in 3-dimensional space. My apologies for the small size and poor visual quality of the video - I am still experimenting with how best to generate animations and to upload them to blogger.
A rotation in 3-dimensional space can be done about an axis pointing in any direction, so there is a whole spherical surface of possible axis directions, which is represented by the sphere in the video. Each point on this spherical surface represents one possible orientation of the rotation axis, and the whole spherical surface is a 2-dimensional manifold representing all of the possible rotation axes. For the purpose of this blog posting only the topology of this manifold is important (i.e. it has a spherical topology); the detailed geometry of the surface is not important.
Once you have decided where the rotation axis points you then rotate about this axis, which generates a 1-dimensional angle subspace that lives outside the 2 angle dimensions that already live within the spherical manifold. Thus far the overall manifold is the original 2-dimensional spherical manifold (generated by the choice of axis direction) times a 1-dimensional manifold (generated by the choice of rotation). If you rotate by one full circuit (i.e. the rotation angle varies from 0 to 2Pi) then you appear to return to the starting point in the 1-dimensional angle subspace, so it seems that this subspace has a standard circular topology. But not all is what it seems to be.
The spherical manifold of possible rotation axes has a hidden degeneracy, because each rotation axis appears twice over. Each point on the sphere and its opposite point (i.e. its antipode) define rotation axes that are parallel, except that they have the opposite physical effect when a given rotation angle is applied about each of these two axes, i.e. the rotation axes are anti-parallel. So a rotation of x about one axis produces the same physical effect as a rotation of -x about its antipodal axis.
How does this degeneracy manifest itself topologically? The additional 1-dimensional angle subspace that is attached to each point on the spherical manifold must be identified with (i.e. glued to) the corresponding 1-dimensional angle subspace that is attached to the antipodal point on the spherical manifold.
This gluing-together of what were hitherto separate parts of the overall manifold imposes the constraint that a rotation of x about one axis produces the same physical effect to a rotation of -x about its antipodal axis. These constraints impose additional topological structure on the original 2-dimensional spherical manifold (generated by axis directions) times a 1-dimensional manifold (generated by rotations), because now antipodal points on the 2-dimensional spherical manifold are glued together via their attached 1-dimensional rotation angle subspaces.
This final glued version of the overall manifold captures all of the topological structure of the group of rotations in 3-dimensional space (i.e. SO(3)). It does not have the same topology as the unglued version because the gluing causes what were hitherto separate points on the manifold to become identified.
This gluing process is illustrated in the video, which shows how the gluing can be implemented by gradually morphing the 1-dimensional manifolds living at antipodal points on the 2-dimensional spherical manifold, until they eventually lie alongside each other ready to be glued together, and then they are finally coalesced into a single 1-dimensional manifold. Colour helps to identify corresponding points on the pair of yet-to-be-glued 1-dimensional manifolds, so that when the colours align with each other then the gluing operation can take place. It is clear from the video that this forces the pair of 1-dimensional manifolds to lie along a diameter of the sphere. Remember that the interior of the sphere is hitherto unused, so it is free to be used to represent rotation angle. Following the original 1-dimensional circular path from a point on the sphere back to itself again now corresponds to following a 1-dimensional path along a diameter of the sphere to its antipodal point, and then instantaneously returning to the starting point which has been glued to its antipodal point.
What consequences does this non-trivial topology have for rotations in 3-dimensional space? A rotation through 2Pi now corresponds to tracing a path along the diameter of the sphere from a point on the surface of the sphere to its antipodal point, and then returning instantaneously to the starting point. There is a non-trivial loop here because the path has clearly travelled somewhere else (i.e. the antipodal point) before returning home, and the loop can't be made to go away by moving it around inside (or on the surface of) the sphere, because the constraint that a pair of points on the path are antipodes is locked into the definition of the path.
So rotating by 2Pi in 3-dimensional space does not return you to where you started, because the path that you follow has a persistence that is enforced by the above antipodal locking-in effect. Of course, the physical object that you rotate has to have an appropriate physical sensitivity to rotation in order to observe this effect. A rotational scalar (e.g. the temperature of an object) will not show this effect, nor will a rotational vector (e.g. directions in 3-dimensional space) show this effect. An example of an object that does show this and other related effects is the Dirac Belt, where an object (i.e. the belt buckle) is rotated whilst the belt itself is used to retain a memory of what rotations have been applied to the buckle. After 2Pi rotation the belt is twisted so the overall state of the system is not the same as before the rotation was applied. The key to making an object that is physically sensitive to a rotation of 2Pi is that it must retain a memory of what rotations have been applied, which is equivalent to the object knowing about the path through the sphere discussed above. Obviously, there are many types of objects that exhibit this 2Pi sensitivity.
The best bit of all is that the locking-in effect of the pair of antipodal points can easily be neutralised by making an additional rotation of 2Pi, so that the overall rotation is 4Pi. This is illustrated in the case of the Dirac Belt, where the twist of the belt is zero after a 4Pi rotation, so the belt is not physically sensitive to rotating its buckle by 4Pi. The reason that the antipodal locking-in effect goes away when you rotate by 4Pi (i.e. make two circuits along a diameter through the sphere in the video) is that there are now two pairs of locked-in antipodal points, which can separately be moved around inside (or on the surface of) the sphere in such a way that the overall length of the loop can be made to go to zero, i.e. equivalent to no path at all. This interesting manipulation is not illustrated in the video.
So the topology of the rotation group SO(3) (i.e. the group of rotations in 3-dimensional space) is non-trivial as shown in the video, and it has highly non-trivial physical consequences such as those illustrated by the Dirac Belt.