On Codimension one Embedding of Simplicial Complexes

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.