On links of vertices in simplicial $d$-complexes embeddable in the euclidean $2d$-space

TL;DR

This paper investigates the properties of $d$-dimensional simplicial complexes embeddable in $2d$-space, establishing new bounds on their size and linking properties of vertex links, with implications for embeddability and linklessness.

Contribution

It introduces a new link property for complexes embeddable in $2d$-space and derives improved upper bounds on the number of $d$-simplices for such complexes.

Findings

Intersection of link complexes is linklessly embeddable in $(2d-1)$-space.

New upper bounds on the number of $d$-simplices in embeddable complexes.

Bounds also apply to linklessly embeddable complexes in $(2d+1)$-space.

Abstract

We consider $d$-dimensional simplicial complexes which can be PL embedded in the $2d$-dimensional euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is linklessly embeddable in the $(2d-1)$-dimensional euclidean space. These considerations lead us to a new upper bound on the total number of $d$-simplices in an embeddable complex in $2d$-space with $n$ vertices, improving known upper bounds, for all $d \geq 2$. Moreover, the bound is also true for the size of $d$-complexes linklessly embeddable in the $(2d+1)$-dimensional space.