I will describe how the geometric topology property of spacetime manifolds and their surgery can constrain quantum statistics data,focusing on 2+1 and 3+1 spacetime dimensions. This gives rise to analogous Verlinde formulas in higher dimensions, presumably useful to bootstrap topological phases of matter. On the other hand, from a different perspective, I will also approach the problem based on lattice models and TQFTs realizable / emergeable in condensed matter. Then we can derive various new link invariants, braiding statistics data (modular S and T matrices) and ground state degeneracy as physical observables to fully characterize those 2+1 and 3+1 dimensional systems. As an example, we can use this tool to one-to-one identify and characterize Z8 classification of Z2xZ2^f-symmetric fermionic Topological Superconductors (realized by stacking layers of a valentine pair of p+ip and p−ip superconductors, where boundary supports non-chiral Majorana-Weyl modes). Work based on: https://arxiv.org/abs/1602.05951 and https://arxiv.org/abs/1612.09298, related 1404.7854 and Refs therein.