Embeddability of join of two simplicial complexes

We say that 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. If two simplicial complexes $K$ and $L$ are not embeddable into double dimension, what can we say about their join complex $K*L$? In this talk, I consider this problem and show that quite unexpectedly there are complexes $K$ and $L$ which are not embeddable into double dimension, but their join is embeddable into double dimension. More generally, I discuss conditions under which the join is non-embeddable into double dimension. The integer Smith classes are our main tool and I discuss these classes and how they behave under the join of complexes with actions of prime period.

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Access code: the Euler characteristic of the wedge of two circles