Hardness of almost embedding simplicial complexes in $\mathbb R^d$

New methods for constructing shellable simplicial complexes

A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}...

Hardness of embedding simplicial complexes in R

Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding ofK into R? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3 (even if k is not considered fixed). We show that the celebrated result of Novikov on the...

Geometric complexity of embeddings in ${\mathbb R}^d$

Given a simplicial complex K , we consider several notions of geometric complexity of embeddings of K in a Euclidean space R : thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any n-complex with N simplices which topologically embeds in R , n > 2, can be PL embedded in R with refinement complexity O(e 4+ǫ ). Families of s...

Cohen-Macaulay-ness in codimension for simplicial complexes and expansion functor

In this paper we show that expansion of a Buchsbaum simplicial complex is $CM_t$, for an optimal integer $tgeq 1$. Also, by imposing extra assumptions on a $CM_t$ simplicial complex, we provethat it can be obtained from a Buchsbaum complex.

On Isomorphism of Simplicial Complexes and Their Related Algebras