Non-embeddability of geometric lattices and buildings

TL;DR

This paper proves that certain complex geometric structures, like thick geometric lattices and finite buildings, cannot be embedded in low-dimensional Euclidean spaces, requiring significantly higher dimensions for embedding.

Contribution

It introduces a new method based on the van Kampen obstruction to establish non-embeddability of order complexes of posets, including geometric lattices and buildings.

Findings

Order complexes of thick geometric lattices require (2d + 1)-dimensional space for embedding.

Finite buildings also exhibit non-embeddability in lower dimensions.

The developed method applies broadly to general order complexes of posets.

Abstract

A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show that order complexes of thick geometric lattices as well as several classes of finite buildings, all of which are order complexes, are hard to embed. That means that such d-dimensional complexes require (2d + 1)-dimensional Euclidean space for an embedding. (This dimension is in general always sufficient for any d-complex.) We develop a method to show non-embeddability for general order complexes of posets which builds on properties of the van Kampen obstruction.