The topology of projective codes and the distribution of zeros of odd maps

TL;DR

This paper explores how the structure of projective codes influences the zeros of odd maps from spheres to Euclidean space, generalizing classical theorems and deriving new bounds and results in topological combinatorics.

Contribution

It generalizes the Borsuk--Ulam theorem, linking code properties to topological results and providing bounds for sphere embeddings and measure partitions.

Findings

New proof of lower bounds for circular chromatic number

Quantitative bounds for sphere embedding genericity

Generalized Ham Sandwich and Borsuk--Shnirel'man theorems

Abstract

We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk--Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean space and show that projective codes give quantitative bounds for a measure of genericity of sphere embeddings; and we prove generalizations of (iii) the Ham Sandwich theorem and (iv) the Lyusternik--Shnirel'man--Borsuk covering theorem for the case where the number of measures or sets in a covering, respectively, may exceed the ambient dimension.