Higher-dimensional book-spaces

TL;DR

This paper constructs, for each dimension n ≥ 2, infinite families of compact simplicial complexes that are topological modular lattices but not topological distributive lattices, answering a question posed by Walter Taylor.

Contribution

It extends Taylor's 2017 result by providing explicit examples of higher-dimensional complexes with the same lattice properties.

Findings

Existence of infinite families of such complexes for all n ≥ 2

Construction of complexes that are modular but not distributive

Answer to Taylor's open question

Abstract

In 2017, Walter Taylor showed that there exist $2$-dimensional simplicial complexes which admit the structure of topological modular lattice but not topological distributive lattice. We give a positive answer to his question as to whether $n$-dimensional simplicial complexes with the same property exist. We do this by giving, for each $n\ge2$, an infinite family of compact simplicial complexes which admit the structure of topological modular lattice but not topological distributive lattice.