Instability of the Smith Index Under Joins and Applications to Embeddability

TL;DR

This paper investigates how the embeddability of simplicial complexes into double dimension is affected by joins, revealing that the Smith index is not stable under joins and providing new conditions based on van Kampen obstructions.

Contribution

It introduces the Smith class as a tool to analyze embeddability, demonstrating that the Smith index can change under joins and establishing new embeddability criteria.

Findings

Existence of complexes that do not embed into double dimension but their joins do

Derived conditions for non-embeddability based on van Kampen obstructions

Showed that the Smith index is not stable under joins

Abstract

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 $\mathbb{Z}_p$-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.