I have not yet had time and opportunity to study QFT in any amount of detail; nonetheless, I have a question that has been on my mind for some time now. Let's say we have a path integral of the very general form

wherein, as usual

Here "D" denotes some generalisation of the integration so that this whole integral becomes diffeomorphism invariant. My questions are :

1. Is it even possible to make a path integral diffeomorphism invariant, or am I going down the wrong track ? Physically, the choice of coordinate basis should have no relevance to the physics encoded in such a path integral, but I'd like to know if it is mathematically possible and rigorous.
2. Assuming diffeomorphism invariance, what happens to a path integral under changes of topology, specifically connectedness ? While choice of coordinates has no physical relevance, making our space-time background multiply connected should change the paths a system can take, and hence the integral. Or not ?
3. Assuming choice of topology even matters in a path integral, is there a way to formulate a path integral such that it is invariant not just under changes in coordinate basis, but also under changes in topology ( again, specifically connectedness ) ?

I am not even sure if the above makes any sense, but this has been on my mind for a while, so I will be grateful for any comments. Note though that on this occasion I am not looking for idle speculation and personal opinions, but rather for technical feedback.