meanwhile, a manifold is a topological space locally homeomorphic to Rd by mappings phi, psi,...which have the following differentiability property: where the domains of two maps overlap,
going from Rd to Rd by the composition of one with the inverse of the other is (either continuously differentiable a certain number of times or) infinitely differentiable.
LQG is usually developed in the d=3 case and the manifold that physically represents space is taken to be "smooth"----which means that the mappings from Rd to Rd which I just mentioned are infinitely differentiable.
LQG can be defined in any dimension d. It is not limited to the d = 3 case, and indeed has been studied in some other cases besides d = 3. But typically the manifold representing space is a compact smooth 3-manifold, a "continuum", denoted by the letter M.
You can get all this from any beginning treatment of LQG like, e.g. Rovelli/Upadhya, or Rovelli/Gaul. Again, if you need links, let me know.
All I have done to supplement the standard treatment that you find there is to define a differentiable manifold. I assume this is very familiar to you Cinquero but some other reader might conceivably want it defined.