Homology of polyomino tilings on flat surfaces

TL;DR

This paper investigates the homology groups associated with polyomino tilings on flat surfaces, providing algebraic tools to prove the impossibility of certain tilings and extending known results to topological surfaces.

Contribution

It introduces a homology-based approach to analyze polyomino tilings on topological surfaces, generalizing planar methods to more complex geometries.

Findings

Homology groups can determine tiling impossibility on certain surfaces.

Several non-existence results for polyomino tilings on square-tiled surfaces.

Extension of planar tiling proofs to topological surface contexts.

Abstract

The homology group of a tiling introduced by M. Reid is studied for certain topological tilings. As in the planar case, for finite square grids on topological surfaces, the method of homology groups, namely the non-triviality of some specific element in the group allows a `coloring proof' of impossibility of a tiling. Several results about the non-existence of polyomino tilings on certain square-tiled surfaces are proved in the paper.