The Euler characteristic of a polyhedral product

TL;DR

This paper presents a straightforward formula for calculating the Euler characteristic of polyhedral products derived from simplicial complexes and pairs of spaces, with simplified versions involving the h-polynomial in special cases.

Contribution

It introduces a simple, explicit formula for the Euler characteristic of polyhedral products, connecting topological invariants with combinatorial data of the underlying simplicial complex.

Findings

Derived a simple formula for the Euler characteristic of polyhedral products.

Simplified the formula in special cases using the h-polynomial.

Established connections between topological and combinatorial invariants.

Abstract

Given a finite simplicial complex L and a collection of pairs of spaces indexed by its vertex set, one can define their polyhedral product. We record a simple formula for its Euler characteristic. In special cases the formula simplifies further to one involving the h-polynomial of L.