A Computational (Co)homological Approach to Contiguity Distance

TL;DR

This paper introduces algebraic (co)homological invariants called (co)homological distances that improve estimates of homotopic distance and demonstrate convergence properties, with explicit computations validating their effectiveness.

Contribution

It defines new algebraic invariants for continuous maps and simplicial maps, providing computable bounds and convergence results that refine classical estimates.

Findings

New invariants provide lower bounds for homotopic distance.

Simplicial cohomological distance converges to continuous cohomological distance after subdivisions.

Explicit computations demonstrate the approach's effectiveness.

Abstract

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.