A higher genus surface can be considered as a connected sum of tori, and the linking cylinder in the connected sum can be replaced by a sum over all the propagating particles. If the theory is modular invariant on the torus, you know that you can do modular transformations on each torus separately, and everything is consistent. It is intuitive that every large diffeomorphism of a high genus surface can be generated using generators each in the individual SL(2,Z)s of the different tori of the connected sum. So by knowing torus modular invariance of the theory, correct for all tori and insertions on the tori, you learn that the large diffeomorphisms are ok.
A sketch of a proof: you cut up the torus along loops into pieces homeomorphic to triangles, then consider the image of the loops under diffeomorphism. Restricting to each torus, there is a nonzero intersection number of the image of the loops with the old loops, and you perform a large diffeomorphism whenever this can reduce the intersection number. continue on each torus until you can't go reduce the intersection numbers any further. At this point, the intersection number must be as low as possible, and this means that the curves are isotopic to their original position, so that the diffeomorphism is now continuously connected to the identity.
I did not check this in detail (I should), but the main lemma you need--- that you can always reduce a non-minimal intersection number by isotoping the curves to go near one genus, and doing an SL(2,Z) is very plausible. There might also be an easier proof.
This post imported from StackExchange Physics at 2014-04-05 17:28 (UCT), posted by SE-user Ron Maimon