Analytic subdivision invariants

TL;DR

This paper develops subdivision invariants based on an inner product on chain complexes, providing tools to analyze and quantify the effects of simplicial complex subdivisions, along with bounds and algorithms for their computation.

Contribution

It introduces a new inner product-based framework for subdivision invariants, along with a combinatorial lower bound and an efficient algorithm for their calculation.

Findings

Invariants do not all vanish for non-trivial subdivisions

A combinatorial lower bound for the invariants is established

An effective algorithm for computing the invariants is provided

Abstract

This paper introduces an inner product on chain complexes of finite simplicial complexes that is well-adapted to the harmonic study of subdivisions. Its definition utilizes a decomposition of the chain spaces that suggests a sequence of subdivision invariants which we show do not all vanish for non-trivial subdivisions. We exhibit a combinatorial lower bound for these invariants and provide an effective algorithm for their computation. Unfortunately, these invariants cannot distinguish every subdivision nor do they necessarily increase over successive subdivisions.