Brieskorn Manifolds, Generalized Sieradski Groups, and Coverings of Lens Spaces

Abstract

A Brieskorn manifold B(p, q, r) is the r-fold cyclic covering of the 3-sphere S³ branched over the torus knot T(p, q). Generalized Sieradski groups S(m, p, q) are groups with an 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 x₁,…, 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 the groups 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 spines of closed 3-manifolds. We establish a similar 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. To classify some of the constructed manifolds, we use Matveev’s computer program “Recognizer.”