Effect of global topology of space on wavefunctions? Usually, we have solve for wavefunctions assuming trivial periodic boundary conditions i.e. we connect, in 2D for example, like a torus. What would be the effect on the spectra or eigenfunctions of a free particle if we non-trivially glued like in one of the ways below?

 A: It makes no sense to ask about "spectra" unless you have an operator whose spectrum you're interested in asking about. An interesting thing you can, however, ask about without reference to an operator, a Hamiltonian for example, would be the bracket algebra. On a topologically non-trivial space, $[x,p]=i\hbar$ can no longer be asserted globally.
However, this is a much more complicated subject, so instead of giving an explanation here, I will instead refer you to the final chapter of the quantum field theory book by Nair. The last chapter is not actually about quantum field theory, but regular quantum mechanics (and classical mechanics with a fair amount of differential geometry in use). One of the subjects of that chapter is how non-trivial phase spaces behave, which would include the situation here as a special case.
Edit: The original question did not specify that OP was interested specifically in the free particle. To this question, let me point out that the role of boundary conditions in QM problems is always to supply a quantization condition on the momentum. Without boundary conditions, any positive energy $E$ could a priori be an eigenvalue of the free Hamiltonian (essentially it's the spectrum on the plane without identifications).
It should be a straightforward exercise to write down the solutions to the eigenvalue problem and impose the identifications (boundary conditions). This will just lead to some quantization conditions on the energy/momentum eigenvalues. Perhaps someone else can comment on the details of this, but I do not plan to work out the details here myself.
