Homology of balanced complexes via the Fourier transform

TL;DR

This paper characterizes the integral coboundaries of subcomplexes of joins of finite abelian groups using Fourier transforms, extending results on topological interpretations of cyclotomic polynomials.

Contribution

It introduces a Fourier transform-based characterization of coboundaries in balanced complexes, linking algebraic topology with harmonic analysis on finite groups.

Findings

Provides a Fourier transform criterion for coboundaries

Extends topological interpretation of cyclotomic polynomials

Connects algebraic topology with harmonic analysis

Abstract

Let G_0,...,G_k be finite abelian groups and let G_0*...*G_k be the join of the 0-dimensional complexes G_i. We give a characterization of the integral k-coboundaries of subcomplexes of G_0*...*G_k in terms of the Fourier transform on the group G_0 \times ... \times G_k. This leads to an extension of a recent result of Musiker and Reiner on a topological interpretation of the cyclotomic polynomial.