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Transactions of the American Mathematical Society

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



Regular coverings of homology $ 3$-spheres by homology $ 3$-spheres

Authors: E. Luft and D. Sjerve
Journal: Trans. Amer. Math. Soc. 311 (1989), 467-481
MSC: Primary 57N10; Secondary 57M10
MathSciNet review: 978365
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Abstract: We study $ 3$-manifolds that are homology $ 3$-spheres and which admit nontrivial regular coverings by homology $ 3$-spheres. Our main theorem establishes a relationship between such coverings and the canonical covering of the $ 3$-sphere $ {S^3}$ onto the dodecahedral space $ {D^3}$. We also give methods for constructing irreducible sufficiently large homology $ 3$-spheres $ \tilde M,\;M$ together with a degree $ 1$ map $ h:M \to {D^3}$ such that $ \tilde M$ is the covering space of $ M$ induced from the universal covering $ {S^3} \to {D^3}$ by means of the degree $ 1$ map $ h:M \to {D^3}$. Finally, we show that if $ p:\tilde M \to M$ is a nontrivial regular covering and $ \tilde M,\;M$ are homology spheres with $ M$ Seifert fibered, then $ \tilde M = {S^3}$ and $ M = {D^3}$.

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Keywords: Homology $ 3$-spheres, coverings, binary icosahedral group, dodecahedral space, degree $ 1$ maps
Article copyright: © Copyright 1989 American Mathematical Society

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