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 Trudy Inst. Mat. i Mekh. UrO RAN, 2017, Volume 23, Number 4, Pages 85–97 (Mi timm1469)

Brieskorn manifolds, generated Sieradski groups, and coverings of lens space

A. Yu. Vesninab, T. A. Kozlovskayac

a Sobolev Institute of Mathematics, Novosibirsk, 630090 Russia
b Novosibirsk State University, Novosibirsk, 630090 Russia

Abstract: The Brieskorn manifold $\mathscr B(p,q,r)$ is the $r$-fold cyclic covering of the three-dimensional sphere $S^{3}$ branched over the torus knot $T(p,q)$. The generalised Sieradski groups $S(m,p,q)$ are groups with $m$-cyclic presentation $G_{m}(w)$, where the word $w$ has a special form depending on $p$ and $q$. In particular, $S(m,3,2)=G_{m}(w)$ is the group with $m$ generators $x_{1},\ldots,x_{m}$ and $m$ defining relations $w(x_{i}, x_{i+1}, x_{i+2})=1$, where $w(x_{i}, x_{i+1}, x_{i+2}) = x_{i} x_{i+2} x_{i+1}^{-1}$. Cyclic presentations of $S(2n,3,2)$ in the form $G_{n}(w)$ were investigated by Howie and Williams, who showed that the $n$-cyclic presentations are geometric, i.e., correspond to the spines of closed three-dimensional manifolds. We establish an analogous result for the groups $S(2n,5,2)$. It is shown that in both cases the manifolds are $n$-fold branched cyclic coverings of lens spaces. For the classification of the constructed manifolds, we use Matveev's computer program “Recognizer.”

Keywords: three-dimensional manifold, Brieskorn manifold, cyclically presented group, Sieradski group, lens space, branched covering.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-07906

DOI: https://doi.org/10.21538/0134-4889-2017-23-4-85-97

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Bibliographic databases:

Document Type: Article
UDC: 514.132+515.162
MSC: 57M05, 20F05, 57M50

Citation: A. Yu. Vesnin, T. A. Kozlovskaya, “Brieskorn manifolds, generated Sieradski groups, and coverings of lens space”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 4, 2017, 85–97

Citation in format AMSBIB
\Bibitem{VesKoz17} \by A.~Yu.~Vesnin, T.~A.~Kozlovskaya \paper Brieskorn manifolds, generated Sieradski groups, and coverings of lens space \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2017 \vol 23 \issue 4 \pages 85--97 \mathnet{http://mi.mathnet.ru/timm1469} \crossref{https://doi.org/10.21538/0134-4889-2017-23-4-85-97} \elib{http://elibrary.ru/item.asp?id=30713962}