Positive Alexander Duality for Pursuit and Evasion

TL;DR

This paper introduces a new positive cohomological criterion for evasion in pursuit-evasion games, providing a necessary and sufficient condition that improves upon previous homological methods by enabling computational solutions.

Contribution

It develops a positive cohomology framework using refined topological tools and Alexander Duality, offering a complete criterion for evasion detection in pursuit-evasion scenarios.

Findings

Positive cohomology criterion is necessary and sufficient for evasion.

The criterion can be computed via linear programming.

Refined topological tools improve evasion analysis.

Abstract

Considered is a class of pursuit-evasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the complement of a coverage region in a Euclidean space over a timeline. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in a general case. The principal tools are (1) a refinement of the Cech cohomology of a coverage region with a positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore homology of the coverage gaps with a positive cone encoding time orientation, and (3) a positive variant of Alexander Duality. Positive cohomology decomposes as the global sections of a sheaf of local positive cohomology over the time axis; we show how this decomposition makes positive cohomology computable as a linear program.