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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: James T. Rogers
Journal: Proc. Amer. Math. Soc. 49 (1975), 232-236
MSC: Primary 54F20
MathSciNet review: 0365521
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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-Čech cohomology with integral coefficients.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0365521-4
PII: S 0002-9939(1975)0365521-4
Keywords: Circle-like, continuum, solenoid, movable, infinite $ P$-height, planar, mapping, Bruschlinsky group, fibration
Article copyright: © Copyright 1975 American Mathematical Society