Non-Embeddability of Geometric Lattices and Buildings

Abstrakt

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 -dimensional complexes require -dimensional Euclidean space for an embedding. (This dimension is always sufficient for any -complex.) We develop a method to show non-embeddability for general order complexes of posets.

Klíčová slova

• Embedding
• Buildings
• Geometric lattice
• Simplicial complex
• Almost embedding
• Finite projective space