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

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] * K with the embedding classes of K. Here [3] * 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 nonzero, then the (m+2)-nd embedding class of [3] * K is non-zero. We also prove some of the consequences of this theorem for the embeddability of [3] * K.

Instability of the Smith Index Under Joins and Applications to Embeddability

“…We remark that this geometric proof works in the metastable dimensions. Unaware of this geometric method, in [16] the author proved this result by an algebraic method using the theory of Smith classes. Although the algebraic method of [16] produces a longer proof than the geometric argument in the case of [3] * N, it can be used to answer our question in more generality, as presented in this paper.…”

“…Method of Proof. We have developed our method "from the ground up" building upon simple observations made first for the case of [3] * K, as explained in [16]. This approach has three main components.…”