On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-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. In addition, we use similar considerations on links of vertices to derive a new asymptotic upper bound on the total number of d-simplices in an (continuously) embeddable complex in 2d-space with n vertices, improving known upper bounds, for all $$d \ge 2$$ . Moreover, we show that the same asymptotic bound also applies to the size of d-complexes linklessly embeddable in the $$(2d+1)$$ -dimensional space.