Polyhedral metrics and 3-manifolds which are virtual bundles. (English) Zbl 0930.57015

Throughout this paper, all 3-manifolds are closed and orientable. Our aim is to give some new examples of 3-manifolds which are virtual bundles, that is, have finite sheeted covers which are surface bundles over the circle. Thurston has raised the question as to whether all irreducible atoroidal 3-manifolds might have this property. The geometrisation conjecture would imply that all such 3-manifolds have hyperbolic metrics, and the virtual bundle question is then equivalent to showing the existence of geometrically infinite incompressible surfaces (compare [W. Thurston, The geometry and topology of 3-manifolds, Lect. Notes Princeton Univ. 1978]).
Our aim is to give several simple constructions of large classes of examples using different polyhedral metrics of non-positive curvature on 3-manifolds (compare [I. R. Aitchison and J. H. Rubinstein, Lond. Math. Soc. Lect. Note Ser. 151, 127-161 (1990; Zbl 0735.57005)]).
Our examples come directly from an observation of Thurston, using different ways of dividing a 3-manifold into simple polyhedra so that there is an overall metric of non-positive curvature. We use some well-known tessellations of Euclidean 3-dimensional space, described in [H. S. M. Coxeter, Proc. Lond. Math. Soc., II. Ser. 43, 33-62 (1937; Zbl 0016.27101)], by truncated octahedra and tetrahedra, plus flying saucers (compare [Aitchison and Rubinstein, loc. cit.]), as well as the easy case of cubes.
In the final section we show that our classes of cubed examples and flying saucers admit geometric decompositions in the sense of Thurston, since they are virtual bundles.


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57R30 Foliations in differential topology; geometric theory
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