A sheaf-theoretic approach to tropical homology

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 Kunneth theorem. Our framework allows for a natural definition of the tropical cycle class map, which we show to be a natural transformation. Finally, we characterize the rational polyhedral spaces that satisfy Poincare-Verdier duality as those that are smooth.& COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we introduce a sheaf-theoretic approach to studying tropical homology. By establishing proper push-forwards and products, we show that tropical homology behaves similarly to classical Borel-Moore homology. We also define the tropical cycle class map and characterize the rational polyhedral spaces that satisfy Poincare-Verdier duality.