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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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.
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