Editing 3D rotation group (section)

==Topology==
{{Main|Hypersphere of rotations}}
The Lie group SO(3) is [[diffeomorphism|diffeomorphic]] to the [[real projective space]] <math>\mathbb{P}^3(\R).</math><ref>{{harvnb|Hall|2015}} Proposition 1.17</ref>

Consider the solid ball in <math>\R^3</math> of radius {{pi}} (that is, all points of <math>\R^3</math> of distance {{pi}} or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotations through an angle {{theta}} between 0 and {{pi}} (not including either) are on the same axis at the same distance. Rotation through angles between 0 and −{{pi}} correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through {{pi}} and through −{{pi}} are the same. So we [[Quotient space (topology)|identify]] (or "glue together") [[antipodal point]]s on the surface of the ball. After this identification, we arrive at a [[topological space]] [[homeomorphic]] to the rotation group.

Indeed, the ball with antipodal surface points identified is a [[smooth manifold]], and this manifold is [[diffeomorphism|diffeomorphic]] to the rotation group. It is also diffeomorphic to the [[real projective space|real 3-dimensional projective space]] <math>\mathbb{P}^3(\R),</math> so the latter can also serve as a topological model for the rotation group.

These identifications illustrate that SO(3) is [[Connected space|connected]] but not [[simply connected]]. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how it is deformed, the start and end point have to remain antipodal, or else the loop will "break open". In terms of rotations, this loop represents a continuous sequence of rotations about the ''z''-axis starting (by example) at the identity (center of the ball), through the south pole, jumping to the north pole and ending again at the identity rotation (i.e., a series of rotation through an angle ''φ'' where ''φ'' runs from 0 to [[turn (geometry)|2{{pi}}]]).

Surprisingly, running through the path twice, i.e., running from the north pole down to the south pole, jumping back to the north pole (using the fact that north and south poles are identified), and then again running from the north pole down to the south pole, so that ''φ'' runs from 0 to 4{{pi}}, gives a closed loop which ''can'' be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The [[plate trick]] and similar tricks demonstrate this practically.

The same argument can be performed in general, and it shows that the [[fundamental group]] of SO(3) is the [[cyclic group]] of order 2 (a fundamental group with two elements). In [[physics]] applications, the non-triviality (more than one element) of the fundamental group allows for the existence of objects known as [[spinor]]s, and is an important tool in the development of the [[spin–statistics theorem]].

The [[universal cover]] of SO(3) is a [[Lie group]] called [[Spin(3)]]. The group Spin(3) is isomorphic to the [[special unitary group]] SU(2); it is also diffeomorphic to the unit [[3-sphere]] ''S''<sup>3</sup> and can be understood as the group of [[versor]]s ([[quaternion]]s with [[absolute value]] 1). The connection between quaternions and rotations, commonly exploited in [[computer graphics]], is explained in [[quaternions and spatial rotation]]s. The map from ''S''<sup>3</sup> onto SO(3) that identifies antipodal points of ''S''<sup>3</sup> is a [[surjective]] [[homomorphism]] of Lie groups, with [[Kernel (algebra)|kernel]] {±1}. Topologically, this map is a two-to-one [[covering map]]. (See the [[plate trick]].)