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The link volume of 3-manifolds is not multiplicative under coverings


Author: Jair Remigio–Juárez
Journal: Proc. Amer. Math. Soc. 144 (2016), 4075-4079
MSC (2010): Primary 57M10, 57M12, 57M25, 57M27
DOI: https://doi.org/10.1090/proc/13008
Published electronically: February 12, 2016
MathSciNet review: 3513562
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Abstract: We obtain an infinite family of $ 3-$manifolds $ \{{M}_n\}_{n\in \mathbb{N}}$ and an infinite family of coverings $ \{\varphi _n:\tilde {M}_n\to M_{n}\}_{n\in \mathbb{N}}$ with covering degrees unbounded and satisfying that $ \mbox {\rm LinkVol}[\tilde {M}]=\mbox {\rm LinkVol}[M].$ This shows that link volume of 3-manifolds is not well behaved under covering maps, in particular, it is not multiplicative, and gives a negative answer to a question posed in a work of Rieck and Yamashita, namely, how good is the bound $ \mbox {\rm LinkVol}[\tilde {M}]\leq q \mbox {\rm LinkVol}[M]$, when $ \tilde {M}$ is a $ q$-fold covering of $ M$?


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Jair Remigio–Juárez
Affiliation: División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, Km. 1 Carr. Cunduacán-Jalpa de Méndezm, Cunduacán, Tab. 86690, Mexico
Email: jair.remigio@ujat.mx

DOI: https://doi.org/10.1090/proc/13008
Received by editor(s): September 24, 2014
Received by editor(s) in revised form: September 8, 2015, and October 21, 2015
Published electronically: February 12, 2016
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2016 American Mathematical Society

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