A sheaf-theoretic approach to tropical homology

TL;DR

This paper develops a sheaf-theoretic framework for tropical homology, enabling functorial properties, duality, and a natural cycle class map, thus bridging tropical and classical homological theories.

Contribution

It introduces a sheaf-theoretic approach to tropical homology, including non-compact supports, and establishes fundamental properties like duality, push-forwards, and product structures.

Findings

Established functorial properties of tropical homology.

Proved Poincaré-Verdier duality over integers on tropical manifolds.

Defined a natural tropical cycle class map.

Abstract

We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves analogously to classical Borel-Moore homology in the sense that there are proper push-forwards, cross products, and cup products with tropical cohomology classes, and that it satisfies identities like the projection formula and the K\"unneth theorem. Our framework allows for a natural definition of the tropical cycle class map, which we show to be a natural transformation. Finally, we prove Poincar\'e-Verdier duality over the integers on tropical manifolds.