Tropical Poincaré duality spaces

Abstract

Certain polyhedral fans can be constructed from matroids, and these serve as the local model of tropical manifolds. Such matroidal fans satisfy a tropical version of Poincaré duality [JRS17]. In this thesis, we give conditions on pure polyhedral fans which are equivalent to this property. Moreover, we classify tropical Poincaré spaces of dimension two. Furthermore, we develop the derived category of cellular sheaves on a polyhedral complex, based on work by Curry [Cur14], and use Verdier duality to prove a vanishing result on the compact support cohomology of the wave sheaf on a Cohen--Macaulay simplicial polyhedral fan.