Tropical Poincar\'e duality spaces

TL;DR

This paper investigates the conditions under which tropical fans exhibit Poincaré duality, providing classifications, necessary conditions, and implications for local duality spaces and abstract polyhedral spaces.

Contribution

It establishes necessary and sufficient conditions for tropical Poincaré duality in fans and introduces classifications and criteria for local duality spaces.

Findings

Classification of tropical Poincaré duality spaces in dimension one

Necessary conditions for fans to be tropical Poincaré duality spaces

Conditions under which local duality implies global duality

Abstract

The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel-Moore homology. When all these cap products are isomorphisms, the fan is said to be a tropical Poincar\'e duality space. If all the stars of faces also are such spaces, such as for fans of matroids, the fan is called a local tropical Poincar\'e duality space. In this article, we first give some necessary conditions for fans to be tropical Poincar\'e duality spaces and a classification in dimension one. Next, we prove that tropical Poincar\'e duality for the stars of all faces of dimension greater than zero and a vanishing condition implies tropical Poincar\'e duality of the fan. This leads to necessary and sufficient conditions for a fan to be a local tropical Poincar\'e duality space. Finally, we use such fans to show that certain abstract balanced polyhedral spaces satisfy tropical Poincar\'e duality.