Fundamental Groups of Simplicial Complexes

TL;DR

This paper introduces two types of simplicial complexes derived from integers and proves their fundamental groups are isomorphic, revealing a deep connection between their topological structures.

Contribution

It defines the common divisor and prime divisor simplicial complexes and establishes an isomorphism between their fundamental groups, a novel topological insight.

Findings

The fundamental groups of the two complexes are isomorphic.

A specific map between the complexes preserves their topological properties.

The complexes are constructed from arbitrary sets of integers.

Abstract

We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a map between the two simplicial complexes, and use this map to show that for any set of integers, the fundamental groups of the resulting simplicial complexes are isomorphic.