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Bredon cohomology methods in mass partition problems on spheres

Surojit Ghosh

Abstract

We apply $\mathrm{RO}(G)$-graded Bredon cohomology to mass assignment problems, extending classical mass partition methods. Within this framework, we reprove a recent result of Lessure and Soberón: for $n+1$ mass assignments on $k$-dimensional affine subspaces of $\mathbb{R}^n$, there exists a $k$-subspace containing a sphere that simultaneously bisects all measures. This approach highlights a flexible topological framework with potential for broader applications.

Bredon cohomology methods in mass partition problems on spheres

Abstract

We apply -graded Bredon cohomology to mass assignment problems, extending classical mass partition methods. Within this framework, we reprove a recent result of Lessure and Soberón: for mass assignments on -dimensional affine subspaces of , there exists a -subspace containing a sphere that simultaneously bisects all measures. This approach highlights a flexible topological framework with potential for broader applications.
Paper Structure (10 sections, 11 theorems, 56 equations)

This paper contains 10 sections, 11 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement.
  3. Preliminaries on Bredon Cohomology
  4. Equivariant formulation
  5. Mass partitions as equivariant maps
  6. An $RO(G)$-graded spectral sequence
  7. Cohomology of certain quotient of $O(n)$
  8. Flag Manifolds over $\mathbb{R}$
  9. The spectral sequence computation
  10. The Proof

Key Result

Proposition 3.1

For a free $G$-CW complex $X$, there exists a spectral sequence with differentials These spectral sequences assemble (as $\alpha$ varies) into a multiplicative $RO(G)$-graded spectral sequence where $s\in\mathbb{Z}$ and $\alpha\in RO(G)$.

Theorems & Definitions (20)

  • Definition 2.1
  • Example 2.2
  • Proposition 3.1
  • Corollary 3.2
  • Remark 3.3
  • Corollary 3.4
  • Proposition 4.1: Bor53
  • Remark 4.2
  • Proposition 4.3
  • Remark 4.4
  • ...and 10 more