One can roughly interpret Betti numbers as the number of n-dimensional holes of our space. Is there a reasonable interpretation for torsion coefficients? Moreover, for 'physical' applications, are they significant?
EDIT: let's use the more accurate vague phrase: "Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces," from wikipedia.
EDIT 2: In a quasi-hamiltonian formulation, the motion of (something) traces out various paths along a manifold which can be modeled as a superposition of a few dynamical systems (at least in this particular situation). We want to simplify the data that describes this motion while preserving some of the 'qualitative' features of the (discrete) topological space. I am certain that when processing the data, we will attempt to preserve the Bhetti numbers. Should I also attempt to preserve the torsion coefficients, if I am to study physical meanings?