A Computational (Co)homological Approach to Contiguity Distance

We introduce two new algebraic invariants, the (co)homological distances between continuous maps, which provide computable lower bounds for the homotopic distance and strictly refine the classical cup-length estimates. We then define the simplicial cohomological distance between simplicial maps and prove a convergence theorem showing that, after sufficiently many barycentric subdivisions, it recovers the cohomological distance between the corresponding continuous maps. Several explicit computations are presented to illustrate the effectiveness of the proposed approach.