Salman Parsa
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Instability of the Smith Index Under Joins and Applications to Embeddability
We say a d-dimensional simplicial complex embeds into double dimension if it embeds into the Euclidean space of dimension 2d. For instance, a graph is planar iff it embeds into double dimension. We study the conditions under which the join of two simplicial complexes embeds into double dimension. Quite unexpectedly, we show that there exist complexes which do not embed into double dimension, however their join embeds into the respective double dimension. We further derive conditions, in terms of the van Kampen obstructions of the two complexes, under which the join will not be embeddable into the double dimension. Our main tool in this study is the definition of the van Kampen obstruction as a Smith class. We determine the Smith classes of the join of two Z_p-complexes in terms of the Smith classes of the factors. We show that in general the Smith index is not stable under joins. This allows us to prove our embeddability results.
On embeddability of joins and their `factors'
with A. Skopenkov
  We present a short and clear proof of the following particular case of a 2006 unpublished result of Melikhov-Schepin. Let K be a k-dimensional simplicial complex and K ∗ [3] the union of three cones over K along their common bases.  If 2d ≥ 3k + 3 and K ∗ [3] embeds into R^(d+2), then K embeds into R^d . We also present a generalization of this theorem. The proofs are based on the HaefligerWeber ‘configuration spaces’ embeddability criterion, equivariant suspension theorem and simple properties of joins and cones

On the Smith classes, the van Kampen obstruction and
 embeddability of [3]∗K 

In this survey-research paper, we first introduce the theory of Smith 
classes of complexes with fixed-point free, periodic maps on them. 
These classes, when defined for the deleted product of a simplicial 
complex K, are the same as the embedding classes of K. Embedding 
classes, in turn, are generalizations of the van Kampen obstruction 
class for embeddability of a d-dimensional complex K into the Euclidean 2d-space. All of these concepts will be introduced in simple 
terms.  Second, we use the theory introduced in the first part to relate the 
embedding classes (or the special Smith classes) of the the complex 
[3] ∗ K with the embedding classes of K. Here [3] ∗ K is the join of K 
with a set of three points. Specifically, we prove that if the m-th embedding class of K is nonzero, then the (m+2)-nd embedding class of [3]∗K is non-zero. We also prove some of the consequences of this theorem for the embeddability of [3] ∗ K.

Bounding the Number of d-Simplices of Embeddable Complexes  
In this research, the objects of study are simplicial complexes that are embeddable into Euclidean space. It is a well-known fact that if a graph (i.e. a 1-d simplicial complex) embeds into the plane, then it can have at most a linear number of edges, in terms of number of vertices. The first higher dimensional analog would be the problem of upper bounding the number of triangles in 2-dimensional simplicial complexes that embed into the four dimensional Euclidean space. It is a conjecture by many, including B. Gruenbaum and G. Kalai that the correct asymptotic upper bound is the number vertices to the power of 2. In a more concice statement the conjecture is that the upper bound is 4 tiems the number of edges. This problem seems far from resolved. 
 
In this work, I have given a critorion that the each intersection of link-complexes of three disjoint vertices of an embeddable complex must satisfy. This triple intersection is a subgraph of the 1-skeleton of given 2-complex. This graph has to be planar. In the paper I have also a weaker general statement for general dimension that is proved by means of elementary algebraic topology.  
This result is published in: 
S. Parsa, “On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space”, Discrete Comput Geom (2017). 
Embedding Homotopy Types of 2-Complexes into 4-Space 
In this research, I have shown by a new proof that each homotopy type of a 2-dimensional simplicial complex has a member simplicial complex that embeds into 4-space. In particular, any finitely presented group appears as a fundamental group of a 2-complex PL embeded into 4-space. These result had been known before, and I have independently proven them. These embeddings can be built in linear time, that implies, knowing that a 2-complex is embeddable in 4-space does not help with regard to efficiency of algorithmic problems related to its fundamental group.  
 
The paper can be downloaded from here: