A Computational (Co)homological Approach to Contiguity Distance

TL;DR

The paper defines computable cohomological and homological distances $\mathrm{H^{\,}D}(f,g;R)$ and $\mathrm{H_{\,}D}(f,g;R)$ as refined lower bounds for the homotopic distance $\mathrm{D}(f,g)$ and shows they improve cup-length bounds via cohomological weights. It introduces a simplicial version $\mathrm{H^{\,}sD}(\varphi,\psi;R)$ and proves an approximation/convergence theorem ensuring that after enough subdivisions this discrete invariant recovers the continuous cohomological distance. The authors demonstrate strict refinements using stable cohomology operations, notably via lens spaces, yielding higher lower bounds for $\mathrm{H^{\,}cat}$ and $\mathrm{H^{\,}TC}$ than classical cup-length. They also provide computational evidence on standard triangulations, establishing a principled, convergent framework for calculating or approximating the contiguity distance in practice.

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.

Figures (1)

• Figure 3.1: A complex $K$ whose geometric realization is contractible but $\mathrm{scat}(\mathrm{sd}^nK)>0$ for every $n\geq0$

Theorems & Definitions (35)

• Definition 1.1
• Remark 1
• Example 1.2
• Remark 2
• Definition 1.3
• Theorem 1.4
• proof
• Theorem 1.5
• proof
• Corollary 1.6