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 $\mathbb{Z}_p$-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.