Necessary Conditions for Geometric Realizability of Simplicial Complexes

TL;DR

This paper establishes a set of linear conditions that must be satisfied for a simplicial complex to be geometrically realizable in Euclidean space, extending previous theoretical results.

Contribution

It introduces a new linear system framework linking simplicial embeddings to integer solutions, broadening understanding of geometric realizability.

Findings

Linear system associated with simplicial complexes

Necessary conditions for embedding in Euclidean space

Extension of Novik's previous work

Abstract

We associate with any simplicial complex $\K$ and any integer $m$ a system of linear equations and inequalities. If $\K$ has a simplicial embedding in $\R^m$ then the system has an integer solution. This result extends the work of I. Novik (2000).