m-Contiguity Distance

In this paper, we systematically develop the m-contiguity distance between simplicial maps as a discrete approximation framework for homotopical complexity in the category of simplicial complexes. We construct an increasing sequence of invariants that approximate the contiguity distance from below. The fundamental properties of m-contiguity distance are established, including invariance under barycentric subdivision, behavior under compositions, and a categorical product inequality. As applications of this theory, we define the m-simplicial Lusternik-Schnirelmann category and the m-discrete topological complexity, proving that each arises naturally as a special case of m-contiguity distance.

💡 Research Summary

The paper introduces a new discrete invariant called the m‑contiguity distance (denoted SDₘ) for simplicial maps between finite simplicial complexes. This invariant is designed as a lower‑approximation scheme for the classical contiguity distance (the discrete analogue of homotopic distance) and, consequently, for homotopical complexity measures such as the Lusternik‑Schnirelmann (LS) category and topological complexity.

Definition and basic properties.
Given simplicial maps ϕ,ψ : K→K′, the m‑contiguity distance SDₘ(ϕ,ψ) is the smallest integer k for which there exists a cover of K by subcomplexes K₀,…,K_k such that for every subcomplex K_j and every simplicial map η : P→K_j from an m‑dimensional simplicial complex P, the compositions ϕ∘η and ψ∘η lie in the same contiguity class. The authors prove several elementary properties (Proposition 3.1): monotonicity under pre‑ and post‑composition, the inequality SDₘ(ϕ,ψ) ≤ SD(ϕ,ψ), and the monotone behavior with respect to the parameter m (if n ≤ m then SDₙ ≤ SDₘ). Consequently the sequence {SDₘ(ϕ,ψ)}ₘ is non‑decreasing and converges to the classical contiguity distance as m→∞ (Theorem 3.4).

m‑simplicial LS‑category.
The authors define the m‑simplicial LS‑category scatₘ(K) as the smallest k such that K can be covered by k + 1 subcomplexes each of which is “m‑categorical”: any map from an m‑dimensional complex into the subcomplex becomes contiguously trivial after inclusion into K. Lemma 3.1 and Theorem 3.2 show that scatₘ(K)=SDₘ(id_K, c_{v₀}) where c_{v₀} is a constant map, establishing that the LS‑category is a special case of the m‑contiguity distance. Moreover scatₘ(K) ≤ scat(K), providing a lower bound for the classical LS‑category.

Stability under barycentric subdivision.
Propositions 3.2 and 3.3 prove that the barycentric subdivision functor sd preserves contiguity classes and respects composition. Using these, Theorem 3.4 establishes the inequality SDₘ(sd(ϕ), sd(ψ)) ≤ SDₘ(ϕ,ψ). Hence repeated subdivision does not increase the m‑contiguity distance, and in the limit (as subdivisions are iterated) the invariant converges to the classical distance.

Categorical product formula.
Because the Cartesian product of simplicial complexes need not be simplicial, the authors work with the categorical product K×L (vertices are ordered pairs, simplices are pairs of simplices). Lemma 4.1 and Theorem 4.1 give a product inequality:
SDₘ(ϕ₁×ϕ₂, ψ₁×ψ₂) ≤ SDₘ(ϕ₁,ψ₁)+SDₘ(ϕ₂,ψ₂).
This mirrors the familiar subadditivity of topological complexity and provides a tool for estimating the invariant on product spaces.

Applications: m‑discrete topological complexity and m‑Schwarz genus.
The paper defines the m‑discrete topological complexity TCₘ(K) as SDₘ(π₁,π₂), where π₁,π₂ are the two projection maps from the Moore path complex PK to K×K. Theorem 6.1 shows that TCₘ(K) coincides with the classical discrete topological complexity when m is large enough, and provides a Schwarz‑type characterization.

Using Moore paths and simplicial fibrations, the authors introduce the m‑dimensional simplicial Schwarz genus and its homotopical counterpart (Theorems 5.2 and 6.3). They prove that for a simplicial fibration the two notions agree, extending the classical Schwarz genus to the discrete, m‑controlled setting.

Overall contribution.
The work establishes a systematic framework that approximates homotopical invariants from below via a discrete parameter m. The m‑contiguity distance unifies several existing discrete invariants, preserves key structural properties (functoriality, subdivision stability, product subadditivity), and yields new families of invariants (m‑LS‑category, m‑discrete TC, m‑Schwarz genus). As m increases, the invariants converge to their classical counterparts, while for small m they provide computationally tractable lower bounds. This bridges continuous homotopy theory with combinatorial topology and opens avenues for practical computation in applied settings such as topological data analysis and network topology.