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On covering dimension and sections of vector bundles

M. C. Crabb

Abstract

An elementary result in point-set topology is used, with knowledge of the mod $2$ cohomology of real projective spaces, to establish classical results of Lebesgue and Knaster-Kuratowski-Mazurkiewicz, as well as the topological central point theorem of Karasev, which is applied to deduce results of Helly-Lovász, Bárány and Tverberg

On covering dimension and sections of vector bundles

Abstract

An elementary result in point-set topology is used, with knowledge of the mod cohomology of real projective spaces, to establish classical results of Lebesgue and Knaster-Kuratowski-Mazurkiewicz, as well as the topological central point theorem of Karasev, which is applied to deduce results of Helly-Lovász, Bárány and Tverberg
Paper Structure (6 sections, 15 theorems, 20 equations)

This paper contains 6 sections, 15 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. The principal result
  3. A theorem of Lebesgue
  4. Karasev's topological central point theorem
  5. Cohomology
  6. Fibrewise joins

Key Result

Theorem 2.1

Let $\xi_1$, $\ldots$, $\xi_n$ be $n$ finite-dimensional real vector bundles over a compact Hausdorff topological space $X$. Suppose that $(U_i)_{i\in I}$ is a finite open cover of $X$ such that each point of $X$ lies in $U_i$ for at most $n$ indices $i\in I$. Suppose that for each $i\in I$ and $k\i

Theorems & Definitions (30)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 20 more