Weighted Simplicial Complexes and Weighted Analytic Torsions

TL;DR

This paper introduces the study of weighted simplicial complexes with real-valued weights, focusing on their weighted homology and analytic torsion, expanding the mathematical framework for such structures.

Contribution

It presents a novel analysis of weighted simplicial complexes with -valued weights, specifically exploring weighted homology and analytic torsion.

Findings

Development of weighted homology theory for -valued weights

Introduction of weighted analytic torsion for these complexes

Potential applications in topological data analysis

Abstract

A weighted simplicial complex is a simplicial complex with values (called weights) on the vertices. In this paper, we consider weighted simplicial complexes with $\mathbb{R}^2$-valued weights. We study the weighted homology and the weighted analytic torsion for such weighted simplicial complexes.