A cohomological characterization of preimages of nonplanar, circle-like continua
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- by James T. Rogers
- Proc. Amer. Math. Soc. 49 (1975), 232-236
- DOI: https://doi.org/10.1090/S0002-9939-1975-0365521-4
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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-Δech cohomology with integral coefficients.References
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Bibliographic Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 49 (1975), 232-236
- MSC: Primary 54F20
- DOI: https://doi.org/10.1090/S0002-9939-1975-0365521-4
- MathSciNet review: 0365521