On Codimension one Embedding of Simplicial Complexes

TL;DR

This paper investigates the conditions under which d-dimensional simplicial complexes can be embedded in (d+1)-dimensional space, providing homological criteria and bounds on their faces.

Contribution

It introduces a homological obstruction for PL embeddability in one higher dimension and offers a systematic method to bound the number of top-dimensional faces.

Findings

Homological condition necessary for embedding

Obstruction criterion for embeddability

Upper bounds on face counts in low dimensions

Abstract

We study $d$-dimensional simplicial complexes that are PL embeddable in $\mathbb{R}^{d+1}$. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such complexes, particularly in low dimensions.