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

Luft, E.; Sjerve, D.

doi:10.1090/S0002-9947-1989-0978365-1

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
DOI: https://doi.org/10.1090/S0002-9947-1989-0978365-1
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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