  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.
