Cellular Sheaf Cohomology in Polymake

Abstract

This chapter provides a guide to our polymake extension cellularSheaves. We first define cellular sheaves on polyhedral complexes in Euclidean space, as well as cosheaves, and their (co)homologies. As motivation, we summarise some results from toric and tropical geometry linking cellular sheaf cohomologies to cohomologies of algebraic varieties. We then give an overview of the structure of the extension cellularSheaves for polymake. Finally, we illustrate the usage of the extension with examples from toric and tropical geometry.

Tropical Poincaré duality spaces

“…We compute its cellular tropical homology and cohomology over Q using the cellular sheaves package [KSW17] for [polymake], we have:…”

Real Tropical Hyperfaces by Patchworking in polymake

“…and ∂(σ ×{z}) = ∂(σ)×{z} defines the boundary maps. In fact this construction is a special case of a cellular (co-)sheaf [11]. Algorithmically it is beneficial that this does not require the geometric construction of RT (f ).…”

Planar tropical cubic curves of any genus, and higher dimensional generalisations

“…In this section we explicitly compute tropical homology of the floor composed surfaces constructed in the proof of Theorem 1.5. We refer to [MZ14,BIMS15,KSW17] for the definition of tropical homology for locally finite polyhedral complexes in the standard projective space TP n . All tropical homology groups are considered with coefficients in R. This section partially generalises results from [Sha13a].…”

“…Homology groups of a tropical variety X are special instances of its tropical homology groups (we refer to [MZ14,BIMS15,KSW17] for the definition of tropical homology for locally finite polyhedral complexes in TP n ). More precisely, the group H j (X; R) is canonically isomorphic to the tropical homology group H 0,j (X; R).…”