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Combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma with a combinatorial proof

Sajal Mukherjee, Pritam Chandra Pramanik

TL;DR

The paper develops a purely combinatorial framework to prove a degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma, avoiding any topology beyond combinatorics. Central to the approach is a combinatorial Hopf trace formula that expresses traces via $\mathcal{V}$-critical and trajectory structures, enabling degree computations without homology. Using this, the authors prove that for joins of $\mathbb{Z}_p$-combinatorial spheres, any $\mathbb{Z}_p$-equivariant map on barycentric subdivisions has degree congruent to 1 modulo $p$, and thus establish a combinatorial degree version of the generalized $\mathbb{Z}_p$-Tucker's lemma. The results provide a robust, purely combinatorial tool for Borsuk--Ulam-type theorems with potential applications in combinatorics and graph theory, and introduce a combinatorial notion of degree that aligns with topological intuition.

Abstract

Combinatorial analogues of classical Borsuk-Ulam-type theorems (e.g., Tucker's lemma, $\mathbb{Z}_p$-Tucker's lemma, etc.) have numerous important applications in combinatorics. In this paper, we formulate a combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma. Our proof is purely combinatorial in the sense that it does not involve homology, cohomology or any other notions from continuous topology. In order to prove the aforementioned degree theorem, as a main technical tool, we prove a Hopf trace-type formula, which is also purely combinatorial and involves no homology. This combinatorial Hopf trace formula is of independent interest.

Combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma with a combinatorial proof

TL;DR

The paper develops a purely combinatorial framework to prove a degree version of a generalized -Tucker's lemma, avoiding any topology beyond combinatorics. Central to the approach is a combinatorial Hopf trace formula that expresses traces via -critical and trajectory structures, enabling degree computations without homology. Using this, the authors prove that for joins of -combinatorial spheres, any -equivariant map on barycentric subdivisions has degree congruent to 1 modulo , and thus establish a combinatorial degree version of the generalized -Tucker's lemma. The results provide a robust, purely combinatorial tool for Borsuk--Ulam-type theorems with potential applications in combinatorics and graph theory, and introduce a combinatorial notion of degree that aligns with topological intuition.

Abstract

Combinatorial analogues of classical Borsuk-Ulam-type theorems (e.g., Tucker's lemma, -Tucker's lemma, etc.) have numerous important applications in combinatorics. In this paper, we formulate a combinatorial degree version of a generalized -Tucker's lemma. Our proof is purely combinatorial in the sense that it does not involve homology, cohomology or any other notions from continuous topology. In order to prove the aforementioned degree theorem, as a main technical tool, we prove a Hopf trace-type formula, which is also purely combinatorial and involves no homology. This combinatorial Hopf trace formula is of independent interest.

Paper Structure

This paper contains 5 sections, 28 theorems, 105 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the combinatorial Hopf trace formula
  4. Combinatorial degree versions of generalized $\mathbb{Z}_p$- Tucker's lemma
  5. Appendix

Key Result

Theorem 1.7

Let $\mathcal{S}$ be a $\mathbb{Z}_p$-combinatorial sphere, and $\operatorname{Bd}^k(\mathcal{S})$ be the $k$-th barycentric subdivision of $\mathcal{S}$. Then, for any $\mathbb{Z}_p$-equivariant simplicial map, $f: \operatorname{Bd}^{k}(\mathcal{S}) \rightarrow \mathcal{S}$,

Figures (4)

  • Figure 1: $\mathcal{K}_1 \cup (x*\mathcal{K}_2)$
  • Figure 2: Commutative diagram corresponding to the chain map $\phi_\#:C_\#(\mathcal{K}, \mathbb{Z})\rightarrow C_\#(\mathcal{K}, \mathbb{Z})$.
  • Figure 3: The combinatorial $1$-sphere $\mathcal{S}$ and gradient vector field $\mathcal{V}$ with $\tau$ and $v_0$ are critical (for example, '$v_i \rightarrow v_{i-1}$' implies $((v_i)^{(0)},(v_iv_{i-1})^{(1)})\in \mathcal{V}$).
  • Figure 4: Commutative diagram corresponding to the chain map $\phi_\#$.

Theorems & Definitions (71)

  • Definition 1.1: $\mathcal{V}$-trajectory
  • Definition 1.2: co-$\mathcal{V}$-trajectory
  • Definition 1.3: Gradient vector field
  • Definition 1.4: Critical simplex
  • Definition 1.5: Critical chain and co-critical chain
  • Definition 1.6: Combinatorial sphere
  • Theorem 1.7
  • Theorem 1.8: Generalized $\mathbb{Z}_p$-Tucker's lemma
  • Theorem 1.9: Combinatorial degree version of generalized $\mathbb{Z}_p$-Tucker's lemma
  • Theorem 1.10: Combinatorial Hopf trace formula
  • ...and 61 more