Polyhedral products associated with polyhedral join products

Abstract

Polyhedral products are a fundamental object in topology, unifying constructions across toric topology, combinatorics and commutative algebra. Determining their homotopy type is a major goal of study of these objects. In this work, we consider how operations on the underlying simplicial complex change the homotopy type of the associated polyhedral product, and the homotopy type of its loop space. There are two major results in this work. The first is a homotopy equivalence for the loop space of the polyhedral product associated with the polyhedral join product, and the second uses this homotopy equivalence to determine when the loop space of a polyhedral product associated with the polyhedral join product has the homotopy type of a product of simply connected spheres and loops on spheres. In the process of proving these results, many combinatorial properties of the polyhedral join product were proven. These properties allow us to construct new simplicial complexes such that the associated polyhedral product satisfies Anick’s conjecture.

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Identifiers

ORCID for Briony Helen Eldridge: orcid.org/0009-0002-9335-7975

ORCID for Stephen Theriault: orcid.org/0000-0002-7729-5527

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