Topology of compact space forms from Platonic solids. II

Cyclic generalizations of two hyperbolic icosahedral manifolds

“…These manifolds are natural generalizations of M 24 in the following sense: M 24 (n), n > 1, is an n-fold strongly-cyclic branched covering of the lens space L 3,1 , branched over the same link as M 24 . The proof of this fact, presented in [3,4], is based on results by Stevens [18] and by Osborne and Stevens [16]. In Theorem 1 we give a purely topological proof of this fact: we consider a Heegaard diagram for the quotient space of M 24 (n) by its cyclic symmetry and we reduce it to the standard genus one Heegaard diagram for L 3,1 .…”

“…Various examples of three-dimensional spherical, Euclidean, or hyperbolic manifolds arise from pairwise isometrical identifications of faces of convex regular polyhedra in corresponding 3-spaces: S 3 , E 3 , or H 3 . The most famous examples are the spherical and hyperbolic dodecahedral manifolds constructed by Weber and Seifert in 1933 [20].…”

“…It was observed by Cavicchioli, Spaggiari and Telloni [3] that the manifolds M 24 and M 25 , arising from the 2π/3-icosahedron, are 3-fold cyclic branched coverings of the lens space L 3,1 branched over some 2-component links. In the present paper we will consider two families of 3-manifolds which are cyclic generalizations of M 24 and M 25 .…”

“…One family of manifolds, namely M 24 (n), n 1, is a generalization of the manifold M 24 from [6]. The manifolds M 24 (n), where M 24 (3) = M 24 , were independently constructed by Cavicchioli, Spaggiari and Telloni [4] for n 3 and by Kozlovskaya [10,11] for n 2. In both cases M 24 (n) was defined via pairwise identifications of the faces of a 3-complex.…”

“…The manifold M 24 (3) is hyperbolic since, by construction, it is obtained from the hyperbolic 2π/3-icosahedron gluing its faces by isometries. It is stated in [4], without proof, that for n > 3 the manifolds M 24 (n) are hyperbolic and a formula for their volumes is given. However the formula turns out to be wrong even for n = 4.…”