Covering with Excess One: Seeing the Topology

TL;DR

This paper explores the topological structure of the space of grid domain coverings with an excess agent, revealing that its topology is homotopy equivalent to a 1-dimensional complex and establishing a formula linking it to the domain's topology.

Contribution

It introduces the topological analysis of covering spaces with excess agents on grid domains, showing their homotopy type and deriving an Euler characteristic formula.

Findings

The topology of the covering space has the homotopy type of a 1D complex.

The Euler characteristic formula relates the topology of the covering space to the domain.

The study applies to grid domains in 2D with N+1 coverings.

Abstract

We have initiated the study of topology of the space of coverings on grid domains. The space has the following constraint: while all the covering agents can move freely (we allow overlapping) on the domain, their union must cover the whole domain. A minimal number $N$ of the covering agents is required for a successful covering of the domain. In this paper, we demonstrate beautiful topological structures of this space on grid domains in 2D with $N+1$ coverings, the topology of the space has the homotopy type of $1$ dimensional complex, regardless of the domain shape. We also present the Euler characteristic formula which connects the topology of the space with that of the domain itself.