For what angles (and why) does the equation for finite rotation fail to work?

Question

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:

1. J. Tseng, Symmetry and Relativity, lecture notes, 2017, page 57-58. The PDF file is available here.

Answer 1

It doesn't fail. The Taylor series for $\exp$ is convergent for every matrix, and the exponential map $\exp: so(d)\to SO(d)$ is surjective, cf. this math.SE post.

The exponential map $\exp$ is not injective, though, as many angle-vectors describe the same rotation.