# The Complement of Polyhedral Product Spaces and the Dual Simplicial Complexes

## TL;DR

This paper introduces complement polyhedral product spaces and dual complexes, explores their properties, and computes their algebraic structures and duality relations, advancing the understanding of polyhedral products in algebraic topology.

## Contribution

It defines and analyzes complement polyhedral product spaces and dual complexes, providing new properties, algebraic computations, and duality results in the context of polyhedral products.

## Key findings

- Established basic properties of complement polyhedral product spaces.
- Computed the universal algebra of polyhedral product complexes under split conditions.
- Proved an Alexander duality isomorphism for certain polyhedral product spaces.

## Abstract

In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral product complexes. Then we compute the universal algebra of polyhedral product complexes under certain split conditions and the Alexander duality isomorphism on certain polyhedral product spaces.