Heaven & hell. (English) Zbl 0737.57004

Aitchison, Iain R.; Rubinstein, J. Hyam

Differential geometry, Proc. 6th Int. Colloq., Santiago de Compostela/Spain 1988, Cursos Congr. Univ. Santiago de Compostela 61, 5-24 (1989).

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[For the entire collection see Zbl 0682.00012.]
This is a paper with an intensely geometric flavor. It is also quite humorous. The title is derived from Escher’s print and there is a discussion of the connection between the theory of three manifolds, the work of Escher, hyperbolic geometry, and the world view of the mystic Emanuel Swedenborg. But the reader should not be mislead by all this; as it is a serious paper with interesting geometric constructions of three manifolds. We shall describe the central construction of the paper. Tesselate the hyperbolic plane with regular right angled hexagons. The edges of the hexagons form geodesics which can be separated into two families, the red and the black. (Is there a connection here with Stendhal?) Each hexagon is alternatingly bordered by red and black edges. Let \(\Gamma\) be the orientation preserving subgroup of the symmetry group of the tesselation that preserves edge color. Torsion free subgroups of \(\Gamma\) are heavenly. Finite index heavenly subgroups of \(\Gamma\) give rise to orientable surfaces, tesselated by hexagons, and with two families of disjoint simple closed curves, the red and black. The surface is thickened and handles are attached, to the red curves on one side and the black on the other, giving a 3-complex which is sometimes a manifold. (There are theorems which tell you precisely when.) The embedded heavenly surface turns out to be a Heegaard surface and (theorem 26) every 3- manifold can be constructed in this way. There are also results on pseudo-Anasov maps, covering spaces, and decomposition of 3-manifolds into Platonic solids. For example (theorem 39) every closed oriented 3- manifold decomposes into icosahedra. This paper will appeal particularly to topologists with a geometric view of the subject.

Reviewer: K.H.Dovermann and M.H.Hilden (Honolulu)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
52C20	Tilings in \(2\) dimensions (aspects of discrete geometry)
51M10	Hyperbolic and elliptic geometries (general) and generalizations
52C22	Tilings in \(n\) dimensions (aspects of discrete geometry)

Keywords:

hyperbolic plane; right angled hexagons; symmetry group; tesselation; orientable surfaces; Heegaard surface; 3-manifold; pseudo-Anasov maps; covering spaces; decomposition of 3-manifolds into Platonic solids; icosahedra

Citations:

Zbl 0682.00012