Tropical singular intersection homology

TL;DR

This paper introduces tropical singular intersection homology theories on rational polyhedral spaces, establishing fundamental properties like Poincaré duality and computing specific cases.

Contribution

It develops new tropical intersection homology theories with duality and pairing structures, expanding the understanding of tropical geometry.

Findings

Established Poincaré duality for non-GM tropical intersection homology.

Defined bilinear pairings analogous to cup and cap products.

Computed homologies in specific cases.

Abstract

We introduce tropical singular intersection homologies (non-GM and GM) with the tropical coefficients on rational polyhedral spaces using their filtrations. We investigate their fundamental properties. In the non-GM case, we give a Poincar\'e duality and two bilinear pairings analogous to the cup and cap products under some assumptions. We compute the homologies in some cases.