Discrete homotopy and homology theories for finite posets

This paper presents a discrete homotopy theory and a discrete homology theory for finite posets. In particular, the discrete and classical homotopy groups of finite posets are always isomorphic. Moreover, this discrete homology theory is related to the discrete homotopy theory through a discrete analogue of the Hurewicz map.

💡 Research Summary

The paper develops a fully discrete homotopy and homology theory for finite posets, establishing a tight correspondence with the classical topological invariants derived from the order complex. Starting from McCord’s theorem, which provides a weak homotopy equivalence between a finite poset X and the geometric realization |K(X)| of its order complex, the authors construct a discrete homotopy group π_Dⁿ(X, x₀) by considering base‑point‑preserving, order‑preserving maps from the integer lattice cube (Zⁿ, ∂Zⁿ) into (X, x₀). The group operation is defined via concatenation of such maps along the first coordinate, and for n ≥ 2 the groups are shown to be abelian.

The central result, Theorem 3.3, proves that the natural map θ: π_Dⁿ(X, x₀) → πₙ(|K(X)|, x₀) is an isomorphism for every n. The proof proceeds by partitioning the geometric n‑cube Lⁿ into subcubes indexed by integer intervals, identifying each subcube with a subgraph γ_T of Zⁿ, and using the homeomorphism λ_T: |K(γ_T)| → Lⁿ. For a continuous map f: Lⁿ → |K(X)| that respects simplices, a quotient space J_f^T(|K(X)|) is built by collapsing along chosen simplex‑wise paths; the induced map e µ_X ∘ J_T(f) : γ_T → X is continuous and order‑preserving, allowing reconstruction of a discrete map whose class maps to the original homotopy class under θ. Injectivity and surjectivity follow from this construction, yielding a complete identification of discrete and classical homotopy groups. An illustrative example computes π₁(S¹) = ℤ using only discrete homotopy, showing a simpler combinatorial approach than traditional covering‑space methods.

In parallel, the authors introduce a discrete cubical homology H^Cube_n(X) based on chains in the integer lattice cubes mapped into X. They define a natural homomorphism ψ: H^Cube_n(X) → H^Simp_n(K(X)) (the usual simplicial homology of the order complex). Theorem 4.7 shows ψ is surjective for a class of “homogeneous” posets where every principal down‑set U_x has the same dimension. However, Example 4.10 demonstrates that ψ need not be injective; in dimension 1 the two homology theories can differ, highlighting that the discrete cubical theory captures finer combinatorial information than classical simplicial homology.

Finally, the paper constructs a discrete Hurewicz map h_D: π_D¹(X, x₀) → H^Cube₁(X). Theorem 5.4 establishes a discrete Hurewicz theorem in dimension 1, proving h_D is an isomorphism and that π_D¹ is precisely the abelianization of H^Cube₁. Theorem 5.5 then proves that h_D coincides exactly with the classical Hurewicz map h: π₁(|K(X)|) → H₁(|K(X)|). Thus the discrete homology theory not only mirrors the classical one but also integrates seamlessly with the discrete homotopy groups.

Overall, the paper delivers three major contributions: (1) a discrete homotopy theory for finite posets whose groups agree with classical homotopy groups in all dimensions; (2) a discrete cubical homology theory linked to, but not always identical with, simplicial homology; and (3) a discrete Hurewicz map that matches the classical one. By translating topological invariants into purely combinatorial language based on Hasse diagrams, the work opens new computational avenues for studying poset‑derived structures, with potential applications in topological data analysis, combinatorial optimization, and the algebraic study of directed graphs.