Covering with Excess One: Seeing the Topology
This work addresses a theoretical problem in computational geometry and topology, offering foundational insights into covering spaces, but it is incremental as it builds on existing studies of coverings.
The paper investigates the topological structure of the space of coverings on 2D grid domains, showing that with N+1 covering agents, the space has the homotopy type of a 1-dimensional complex, independent of domain shape, and provides an Euler characteristic formula linking the topology of the covering space to the domain.
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.