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Frontmatter

Chapter One. S1-Bundles Over Surfaces

Abstract
A closed surface is a compact, connected 2-manifold without boundary. From the tangent bundle TX of a closed surface X we can construct the spherical (or unit) tangent bundle of X, denoted ST(X), as the subbundle of TX consisting of vectors of norm 1 (see [GP], page 55). The fiber of ST(X) is the 1-sphere S1, and thus ST(X) is a closed 3-manifold, which always has a canonical orientation, even when the base X of the bundle is non-orientable (see [GP], pp. 76 and 106). These S1-bundles were known by the suggestive name of “bundles of oriented line elements” (see [ST]). The “bundles of unoriented line elements” of X, denoted by PT(X), are obtained from ST(X) by identifying the vectors (x, v) and (x, −v), for every (x, v) E ST(X). The S1-bundle PT(X), which is also called projective tangent bundle, is a canonically oriented, closed 3-manifold and the natural map ST(X)→PT(X) is a 2-fold covering.
José María Montesinos-Amilibia

Chapter Two. Manifolds of Tessellations on the Euclidean Plane

Abstract
The space whose points are the different positions of a regular dodecahedron inscribed in the 2-sphere, with the most natural possible topology, is a closed, orientable 3-manifold known as the Poincaré homology 3-sphere A dodecahedron is a tessellation of the 2-sphere, as is an octahedron or a tetrahedron, for instance. The original examples of tessellations belong to the euclidean plane ℝ2, like the hexagonal mosaics that one can admire in The Alhambra de Granada or in The Aljaferia de Zaragoza The hyperbolic plane H2 is very rich in tessellations. The object of the rest of the book is to describe the 3-manifolds of euclidean, spherical and hyperbolic tessellations.
José María Montesinos-Amilibia

Chapter Three. Manifolds of Spherical Tessellations

Abstract
The spherical tessellations are represented by the platonic solids or regular polyhedra. Thus the set of positions of a platonic solid inscribed in the 2-sphere S2 is a closed 3-manifold. The fundamental group of such a manifold is an extension of ℤ2 by the group of isometries of the platonic solid. The subgroup ℤ2 is the center of the group. The manifold can be thought of as the spherical tangent bundle of a 2-dimensional spherical orbifold, or as a 3-dimensional spherical orbifold, i.e. the quotient of S3 under a finite subgroup of SO(4). Some material of this chapter was borrowed in part from [DuV].
José María Montesinos-Amilibia

Chapter Four. Seifert Manifolds

Abstract
The spherical bundles of surfaces as well as the manifolds of tessellations are examples of Seifert manifolds. Using the language of orbifolds, a Seifert manifold is a manifold (i.e. an orbifold with empty singular set) that fibers over a 2-orbifold whose singular points form a discrete set (and have cyclic isotropy groups). The manifolds of tessellations are the spherical bundles of such 2-orbifolds.
José María Montesinos-Amilibia

Chapter Five. Manifolds of Hyperbolic Tessellations

Abstract
In this short chapter we describe the hyperbolic tessellations and the 3-manifolds that they define. In contrast to the euclidean and spherical cases we will only give the Seifert invariants of the manifolds of hyperbolic tessellations, and we will not describe them as polyhedra with identified faces.
José María Montesinos-Amilibia

Errata

Without Abstract
José María Montesinos-Amilibia

Backmatter

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