Topology and Quantum mechanics I have a very simple question. Can we know about the topology of the underlying space-time manifolds from Quantum mechanics calculations? If the Space-time is not simply connected, how can one measure the effects of this from calculating transition amplitudes in the path integral formalism?
 A: Let us for simplicity consider non-relativistic quantum mechanics of a particle on a spatial, $3$-dimensional, connected (possibly curved, possibly non-simply connected)  manifold $M$ in the path integral formalism. Then the Feynman propagator 
$$\tag{1} K({\bf q}_f, t_f; {\bf q}_i, t_i)~\sim ~ \int_{{\bf q}(t_i)={\bf q}_i}^{{\bf q}(t_f)={\bf q}_f} \!{\cal D}{\bf q}~ e^{\frac{i}{\hbar}S[q]} $$   
in principle carries detailed information about the full manifold $M$, since we should sum over all histories in the path integral (1). For instance, if the manifold $M$ is not simply connected, then the overlap (1) will depend on path-histories that are not homotopic. 
How the information about the manifold $M$ can be extracted from only knowing the Feynman propagator $K({\bf q}_f, t_f; {\bf q}_i, t_i)$ is an example of an inverse scattering problem.$^1$ 
--
$^1$ Note that these hearing-the-shape-of-a-drum-type problems may not always have unique solutions for the manifold $M$.
A: Here is an experimentalist's answer:
One is continually checking calculations, using the full Quantum Mechanical toolkit,  with the   the data, looking for deviations as hints for new physics beyond the Standard Model. The g-2 of the muon , for example, starts showing some difference between data and theory, not yet statistically significant. 
The standard theoretical toolkit assumes simple connections for space time. Thus, any deviations from "classical" calculations are open to interpretation by new theories, which could include not simply connected space time. These new models would have to incorporate all the present agreements and justify the putative new deviations from the SM.
Here I am answering your " Can we know " question. If we see deviations from the classical calculation we can know that something has to give. If it is connectivity of space time somebody has to calculate its effect.
