A.Yu. Vesnin, T.A. Kozlovskaya. Brieskorn manifolds, generated Sieradski groups, and coverings of lens space ... С. 85-97

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.

The paper was received by the Editorial Office on August 7, 2017.

Andrei Yur’evich Vesnin, Dr. Phys.-Math. Sci, RAS Corresponding Member, Prof., Sobolev Institute of Mathematics, Novosibirsk, 630090 Russia; Novosibirsk State University, Novosibirsk, 630090 Russia,
e-mail: vesnin@math.nsc.ru

Tat’yana Anatol’evna Kozlovskaya, Cand. Sci. (Phys.-Math.), Magadan Institute of Economics, Magadan, 685000 Russia,
e-mail: konus_magadan@mail.ru


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