Restricting cohomology classes to disk and segment configuration spaces

TL;DR

This paper investigates how the cohomology of configuration spaces of disks and segments inside a disk varies with radius, focusing on when certain classes vanish or when continuous sections exist, presenting partial results and open questions.

Contribution

It introduces new questions and partial results on the dependence of cohomology and section existence on the radius in disk and segment configuration spaces.

Findings

Identifies conditions for cohomology class restriction to vanish.

Determines when the angle map admits a continuous section.

Provides numerous conjectures and open problems.

Abstract

The configuration space of n labeled disks of radius r inside the unit disk is denoted Conf_{n, r}(D^2). We study how the cohomology of this space depends on r. In particular, given a cohomology class of Conf_{n, 0}(D^2), for which r does its restriction to Conf_{n, r}(D^2) vanish? A related question: given the configuration space Seg_{n, r}(D^2) of n labeled, oriented segments of length r, it has a map to (S^1)^n that records the direction of each segment. For which r does this angle map have a continuous section? The paper consists of a collection of partial results, and it contains many questions and conjectures.