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Balanced sets and homotopy invariants of covers

Mikhail V. Bludov

TL;DR

The work develops a cohesive homotopy-theoretic framework for covers defined relative to a balanced pair $(V,r)$, linking combinatorial balanced subsets to topological invariants. It proves that the non-balanced complex $\mathcal{K}(V,r)$ has sphere-type homotopy when $r$ lies in the interior of $\mathrm{conv}(V)$, and that the homotopy class of a cover is determined up to a $\mathbb{Z}_2$-involution by the balanced-equivalence class of $(V,r)$. The authors extend cover-extension results to the balanced setting, define a degree and a balanced-intersection index for covers of Euclidean space, and derive KKMS, Sperner, and KKM lemmas as corollaries, providing a versatile toolkit for discrete fixed-point type results across Euclidean and manifold contexts.

Abstract

In this paper, we study a construction of homotopy invariants of open or closed covers, where the homotopy class is defined relative to a pair $(V,r)$, with $V$ a finite set of points in $\mathbb{R}^d$ and $r$ a point in the interior of their convex hull. We show that the simplicial complex of non-balanced subsets associated with $(V,r)$ has the homotopy type of a sphere, and use this to develop a theory of homotopy invariants of covers relative to balanced sets. A key result is that the homotopy class of a cover depends only, up to an involution, on the balanced-equivalence class of $(V,r)$. As applications, we obtain extension theorems for covers in this setting and derive the KKMS lemma, its analogues, and related combinatorial fixed-point results.

Balanced sets and homotopy invariants of covers

TL;DR

The work develops a cohesive homotopy-theoretic framework for covers defined relative to a balanced pair , linking combinatorial balanced subsets to topological invariants. It proves that the non-balanced complex has sphere-type homotopy when lies in the interior of , and that the homotopy class of a cover is determined up to a -involution by the balanced-equivalence class of . The authors extend cover-extension results to the balanced setting, define a degree and a balanced-intersection index for covers of Euclidean space, and derive KKMS, Sperner, and KKM lemmas as corollaries, providing a versatile toolkit for discrete fixed-point type results across Euclidean and manifold contexts.

Abstract

In this paper, we study a construction of homotopy invariants of open or closed covers, where the homotopy class is defined relative to a pair , with a finite set of points in and a point in the interior of their convex hull. We show that the simplicial complex of non-balanced subsets associated with has the homotopy type of a sphere, and use this to develop a theory of homotopy invariants of covers relative to balanced sets. A key result is that the homotopy class of a cover depends only, up to an involution, on the balanced-equivalence class of . As applications, we obtain extension theorems for covers in this setting and derive the KKMS lemma, its analogues, and related combinatorial fixed-point results.
Paper Structure (6 sections, 22 theorems, 38 equations)

This paper contains 6 sections, 22 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Homotopy classes of covers
  3. Balanced subsets
  4. Balanced subsets and covers
  5. Covers of Euclidean space
  6. KKMS and Sperner's lemmas as consequences of Theorem \ref{['VExtSph']}

Key Result

Theorem 1

Let $r$ be a point from the interior of $\mathop{\rm conv}\nolimits(V)$. Then $|\mathcal{K}(V,r)|$ is homotopy equivalent to a sphere $S^{d-1}$

Theorems & Definitions (58)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1: Definition 1.1. from MusH
  • Example 1
  • Example 2
  • Definition 2.2: Definition 1.2 from MusH
  • Remark 3
  • Lemma 2.1: Lemma 1.4 from MusH
  • ...and 48 more