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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Decompositions of continua over the hyperbolic plane


Author: James T. Rogers
Journal: Trans. Amer. Math. Soc. 310 (1988), 277-291
MSC: Primary 54F20; Secondary 54B15, 54F50
MathSciNet review: 965753
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Abstract: The following theorem is proved.

Theorem. Let $ X$ be a homogeneous continuum such that $ {H^1}(X) \ne 0$. If $ \mathcal{G}$ is the collection of maximal terminal proper subcontinua of $ X$, then

(1) The collection $ \mathcal{G}$ is a monotone, continuous, terminal decomposition of $ X$,

(2) The nondegenerate elements of $ \mathcal{G}$ are mutually homeomorphic, indecomposable, cell-like, terminal, homogeneous continua of the same dimension as $ X$,

(3) The quotient space is a homogeneous continuum, and

(4) The quotient space does not contain any proper, nondegenerate, terminal subcontinuum.

This theorem is related to the Jones' Aposyndetic Decomposition Theorem. The proof involves the hyperbolic plane and a subset of the circle at $ \infty $, called the set of ends of a component of the universal cover of $ X$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0965753-1
PII: S 0002-9947(1988)0965753-1
Keywords: Continuum, curve, homogeneous, Effros property, aposyndetic decomposition, terminal subcontinuum, hyperbolic plane
Article copyright: © Copyright 1988 American Mathematical Society