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Revisiting the Nandakumar-Ramana Rao Conjecture

Surojit Ghosh, Ankit Kumar

TL;DR

This work proves the generalized Nandakumar-Ramana Rao conjecture for prime $p$ by leveraging $RO(C_p)$-graded Bredon cohomology of configuration spaces, treated as a module over the $RO(C_p)$-graded cohomology of a point. The authors compute the $bM_p$-module structure on the cohomology of the universal space $EC_p$ and of the configuration space $ ext{Conf}_p(R^d)$, and analyze the action of Euler-type generators to set up obstruction theory. The core contribution is a non-existence result for a $C_p$-equivariant map from $ ext{Conf}_p(R^d)$ to a representation sphere, which implies the partition into $p$ equal-measure convex pieces with additional constraints. This approach provides a novel, algebraic-topological route to measure-partition problems, distinct from index-theoretic methods, and highlights the utility of RO( $C_p$ )-graded cohomology in equivariant combinatorial geometry.

Abstract

We reprove the generalized Nandakumar-Ramana Rao conjecture for the prime case using representation ring-graded Bredon cohomology. Our approach relies solely on the $RO(C_p)$-graded cohomology of configuration spaces, viewed as a module over the $RO(C_p)$-graded Bredon cohomology of a point.

Revisiting the Nandakumar-Ramana Rao Conjecture

TL;DR

This work proves the generalized Nandakumar-Ramana Rao conjecture for prime by leveraging -graded Bredon cohomology of configuration spaces, treated as a module over the -graded cohomology of a point. The authors compute the -module structure on the cohomology of the universal space and of the configuration space , and analyze the action of Euler-type generators to set up obstruction theory. The core contribution is a non-existence result for a -equivariant map from to a representation sphere, which implies the partition into equal-measure convex pieces with additional constraints. This approach provides a novel, algebraic-topological route to measure-partition problems, distinct from index-theoretic methods, and highlights the utility of RO( )-graded cohomology in equivariant combinatorial geometry.

Abstract

We reprove the generalized Nandakumar-Ramana Rao conjecture for the prime case using representation ring-graded Bredon cohomology. Our approach relies solely on the -graded cohomology of configuration spaces, viewed as a module over the -graded Bredon cohomology of a point.

Paper Structure

This paper contains 8 sections, 10 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement.
  3. Preliminaries on Bredon cohomology
  4. Some $\mathbb{M}_{p}$-modules
  5. Connection with equivariant maps and the proof
  6. The equivariant map
  7. Bredon cohomology of representation spheres
  8. The proof

Key Result

Proposition 3.1

There is a spectral sequence with boundary $d_r : E_r^{s,t}(\alpha) \to E_r^{s+r,t-r+1}(\alpha).$ The spectral sequences assemble together for various $\alpha$ into a multiplicative $RO(C_p)$-graded spectral sequence where $s\in \mathbb Z$ and $\alpha \in RO(G)$.

Theorems & Definitions (19)

  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Proposition 3.1
  • Corollary 3.2
  • Remark 3.3
  • Corollary 3.4
  • Proposition 3.6: BG21, Proposition 3.6
  • Proposition 3.7: BG21, Proposition 3.10
  • Proposition 3.8
  • ...and 9 more