The Realization of 3D Topological Spaces Branched Over Graphs

TL;DR

This paper introduces an algorithm to automatically determine 3D topological spaces using branched coverings over graphs, with applications in quantum gravity and spacetime topology visualization.

Contribution

It presents a novel computational method for analyzing 3D topological spaces via branched coverings, including new visualizations to enhance understanding in quantum gravity contexts.

Findings

Successfully identified topological spaces branched over graphs in 3D

Developed visualizations to clarify branched covering structures

Applied algorithm to simple 3D examples

Abstract

In this paper we present an implementation of a computer algorithm that automatically determines the topological structure of spacetime, using a branched covering space representation. This algorithm is applied to a few simple examples in dimension 3, and a complete set of the topological spaces branched over several graphs are found. We also include some new visualizations of the branched covering construction, in order to aid and clarify the understanding of how these structures can be used in quantum gravity to realize the topological nature of the spacetime foam.