Complement Spaces, Dual Complexes and Polyhedral Product Spaces

TL;DR

This paper introduces and analyzes complement polyhedral product spaces, dual complexes, and polyhedral join complexes, establishing their fundamental properties and algebraic structures, including universal algebra and Alexander duality under specific conditions.

Contribution

It defines new classes of polyhedral spaces and computes their algebraic invariants, extending the understanding of duality and join operations in polyhedral topology.

Findings

Established properties of complement polyhedral product spaces

Computed universal algebra of polyhedral join complexes under split conditions

Derived 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 join complexes. Then we compute the universal algebra of polyhedral join complexes under certain split conditions and the Alexander duality isomorphism on certain polyhedral product spaces.