Evasion paths by homotopy limits

TL;DR

This paper investigates the conditions under which evasion paths exist in dynamic spaces using homotopy theory, providing a complete calculation for certain spaces and a new characterization based on homotopy limits.

Contribution

It introduces a novel application of the Bousfield-Kan spectral sequence to analyze evasion paths and characterizes '0-sectional' spaces where these paths can be fully computed.

Findings

Complete calculation of evasion paths for '0-sectional' spaces

Characterization of spaces where evasion paths are computable

Reframing of calculations in terms of covered regions

Abstract

The evasion paths problem asks when a dynamically changing space can be navigated: imagine guards are patrolling a region, for instance, and we need to stay outside their view. We use the Bousfield-Kan spectral sequence for homotopy inverse limits as a proxy for calculating the homotopy groups of the space of evasion paths. This gives a complete calculation of the number of paths (up to homotopy) for spaces we call ``0-sectional." We give a characterization such spaces, and conclude the paper by reframing the calculations in terms of the covered region.