A cohomological characterization of preimages of nonplanar, circle-like continua

Abstract:

Let $G$ be an additively written Abelian group, and let $P = \{ {p_1},{p_2},{p_3}, \cdots \}$ be a sequence of positive integers. An element $g$ in $G$ is said to have infinite $P$-height if (1) $g \ne 0$, (2) each ${p_i} > 1$, and (3) for each positive integer $n$, there is an element $h$ in $G$ such that $({p_1}{p_2} \cdots {p_n})h = g$. The purpose of this paper is to prove the following Theorem. If $X$ is a continuum, then the following are equivalent: (1) ${H^1}(X)$ contains an element of infinite $P$-height, for some sequence $P$ of positive integers; (2) $X$ can be mapped onto a solenoid; (3) $X$ can be mapped onto a nonplanar, circle-like continuum. Here ${H^1}(X)$ is Alexander-Čech cohomology with integral coefficients.