Positive Alexander Duality for Pursuit and Evasion

Robert Ghrist; Sanjeevi Krishnan

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arXiv · 1507.04741

Positive Alexander Duality for Pursuit and Evasion

Robert Ghrist·Sanjeevi Krishnan

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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.

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Robert Ghrist, Sanjeevi Krishnan. 2015-07-16. Positive Alexander Duality for Pursuit and Evasion. https://arxiv.org/abs/1507.04741

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