Positive Alexander Duality for Pursuit and Evasion

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 time. 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 the general case. The principal tools are (1) a refinement of the Čech 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 using techniques of computational polyhedral geometry and linear programming.