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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 <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$3$">-spheres, coverings, binary icosahedral group, dodecahedral space, degree <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$1$"> maps
Article copyright: © Copyright 1989 American Mathematical Society