Must be the wave function single-valued? Ring vs. periodic boundary conditions, spinors and double rotations

This is a lecture notes for undergraduate students. We try to tackle the single valuedness of spatial and double valuedness of spin functions. Also, we adress the need of spinors to accommodate spin functions with some parallelism to the need of axial vectors (or antisymmetric traceless tensors) to accommodate angular momentum. Finally, we revisit the Dirac and Weyl tricks on the non-equivalence of a 2 pi and a 4 pi rotation related the topology of rotation and unitary groups.