## 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

- R. H. Bing,
*Embedding circle-like continua in the plane*, Canadian J. Math.**14**(1962), 113β128. MR**131865**, DOI 10.4153/CJM-1962-009-3 - R. H. Bing,
*Snake-like continua*, Duke Math. J.**18**(1951), 653β663. MR**43450** - M. K. Fort Jr.,
*Images of plane continua*, Amer. J. Math.**81**(1959), 541β546. MR**106441**, DOI 10.2307/2372912 - W. T. Ingram,
*Concerning non-planar circle-like continua*, Canadian J. Math.**19**(1967), 242β250. MR**214037**, DOI 10.4153/CJM-1967-016-3 - George W. Henderson,
*Continua which cannot be mapped onto any non planar circle-like continuum*, Colloq. Math.**23**(1971), 241β243, 326. MR**307203**, DOI 10.4064/cm-23-2-241-243 - James Keesling,
*An algebraic property of the Δech cohomology groups which prevents local connectivity and movability*, Trans. Amer. Math. Soc.**190**(1974), 151β162. MR**367973**, DOI 10.1090/S0002-9947-1974-0367973-6 - L. S. Pontryagin,
*Topological groups*, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR**0201557** - James T. Rogers Jr.,
*The pseudo-circle is not homogeneous*, Trans. Amer. Math. Soc.**148**(1970), 417β428. MR**256362**, DOI 10.1090/S0002-9947-1970-0256362-7 - James Ted Rogers Jr.,
*Pseudo-circles and universal circularly chainable continua*, Illinois J. Math.**14**(1970), 222β237. MR**264622** - Edwin H. Spanier,
*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR**0210112**

## 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