I am learning rotations and group theory/representations and my lecturer's note mentioned that:
"The group is considered connected, but not simply connected... As a result, the formula for a finite rotation, R = $e^{−iθ·J}$ doesn’t work for all angles."
May I ask for what angles (and why) does the above equation fail to work?
References:
J. Tseng, Symmetry and Relativity, lecture notes, 2017. The PDF file is available here (page 57-58): http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures.pdf
I'd call that a case of bad formulation leading to confusion in the reader as to what group is actually discussed.. The chapter starts with "... think of the rotations in 3D of a rigid body ..." but later assumes that spacial inversion $(x,y,z)\mapsto(-x,-y,-z)$ is also in his (otherwise not clearly specified) group. If we deal with movements we can perform with a rigid object, then spacial inversion is not among them. If we deal with linear transformations that leave the given rigid body (a block of wood aka. a parallelepiped) fixed, "infinitesimal" rotations are not among them.
What the author is up to, is the group $O(3)$ of orthogonal transformations n three dimensions, i.e., all linear transformations that respect lengths and angles. This group indeed has two connected components: the subgroup $SO(3)$ is "special" orthogonal transformations, the orthogonal transformations that additionally leave orientation intact (and these can be described by a rotation around an axis), and the complement (which is of course by itself not a group) of such transformations that flip the orientation (which a rotation cannot do).