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Borsuk-Ulam type theorem for Stiefel manifolds and orthogonal mass partitions

Oleg R. Musin

Abstract

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee--for a given set of $m$ measures in $\mathbb{R}^d$--the existence of $k$ mutually orthogonal hyperplanes, any $n$ of which partition each of the measures into $2^n$ equal parts. If $n=k$, the result corresponds to the bound obtained in [11], but with the stronger conclusion that the hyperplanes are mutually orthogonal.

Borsuk-Ulam type theorem for Stiefel manifolds and orthogonal mass partitions

Abstract

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on that guarantee--for a given set of measures in --the existence of mutually orthogonal hyperplanes, any of which partition each of the measures into equal parts. If , the result corresponds to the bound obtained in [11], but with the stronger conclusion that the hyperplanes are mutually orthogonal.
Paper Structure (14 sections, 25 theorems, 85 equations)

This paper contains 14 sections, 25 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Borsuk--Ulam theorems for spheres and Stiefel manifolds.
  3. Orthogonal mass partition.
  4. Borsuk--Ulam type theorems
  5. $G$--BUT manifolds and equivariant cobordisms.
  6. BUT -- theorem for $G={\Bbb Z}/2$.
  7. $({\Bbb Z}/2)^k$--cobordisms.
  8. General BUT -- theorem for $G=({\Bbb Z}/2)^k$.
  9. Borsuk--Ulam theorem for the product of spheres.
  10. Borsuk--Ulam type theorems for Stiefel manifolds.
  11. Equipartition of measures: functions $g_\mu$
  12. Hyperplanes in $\mathbb R^d$
  13. Function $g_\mu$.
  14. Proof of the theorem on orthogonal partitions

Key Result

Theorem 1.1

Let $q_\ell=\varepsilon_{\ell,1}\lambda_1+...+\varepsilon_{\ell,k}\lambda_{k}, \, \ell=1,...,m$, be elements of $G=({\Bbb Z}/2)^k$. Suppose Then for any continuous equivariant mapping the zeros set $Z_f:=f^{-1}(0)$ is non-empty.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • ...and 25 more