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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Cloverleaf representations of simply connected $ 3$-manifolds


Author: Edwin E. Moise
Journal: Trans. Amer. Math. Soc. 201 (1975), 1-30
MSC: Primary 57C05; Secondary 57A10, 57C40
DOI: https://doi.org/10.1090/S0002-9947-1975-0350745-7
MathSciNet review: 0350745
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Abstract: Let $ M$ be a triangulated $ 3$-manifold satisfying the hypothesis of the Poincaré Conjecture. In the present paper it is shown that there is a finite linear graph $ {K_1}$ in the $ 3$-sphere, with exactly two components, and a finite linear graph $ {K_2}$ in $ M$, such that when the components of the graphs $ {K_i}$ are regarded as points, the resulting hyperspaces are homeomorphic. $ {K_2}$ satisfies certain conditions which imply that each component of $ {K_2}$ is contractible in $ M$. Thus the conclusion of the theorem proved here is equivalent to the hypothesis of the Poincaré Conjecture.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0350745-7
Keywords: Hyperspaces of the $ 3$-sphere, Poincaré Conjecture
Article copyright: © Copyright 1975 American Mathematical Society