On the Smith classes, the van Kampen obstruction and embeddability of $[3]\ast K$

Abstract

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 $[3]\ast K$ with the embedding classes of $K$. Here $[3]\ast K$ is the join of $K$ with a set of three points. Specifically, we prove that if the $m$-th embedding class of $K$ is non-zero, then the $(m+2)$-nd embedding class of $[3]\ast K$ is non-zero. We also prove some of the consequences of this theorem for the embeddability of $[3]\ast K$.