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Some footnotes on Thurston's Notes ''The Geometry and Topology of 3-manifolds''

Athanase Papadopoulos

Abstract

These are a few historical remarks, addenda and references with comments on some topics discussed by Thurston in his notes ''The geometry and topology of three-manifolds''. The topics are mainly hyperbolic geometry, geometric structures, volumes of hyperbolic polyhedra and the so-called Koebe--Andreev--Thurston theorem. I discuss in particular some works of Lobachevsky, Andreev and Milnor, with an excursus in Dante's cosmology, based on the insight of Pavel Florensky. The final version of this paper will appear as a chapter in the book ''In the tradition of Thurston. III'', ed. K. Ohshika and A. Papadopoulos, Springer Nature Switzerland, 2024.

Some footnotes on Thurston's Notes ''The Geometry and Topology of 3-manifolds''

Abstract

These are a few historical remarks, addenda and references with comments on some topics discussed by Thurston in his notes ''The geometry and topology of three-manifolds''. The topics are mainly hyperbolic geometry, geometric structures, volumes of hyperbolic polyhedra and the so-called Koebe--Andreev--Thurston theorem. I discuss in particular some works of Lobachevsky, Andreev and Milnor, with an excursus in Dante's cosmology, based on the insight of Pavel Florensky. The final version of this paper will appear as a chapter in the book ''In the tradition of Thurston. III'', ed. K. Ohshika and A. Papadopoulos, Springer Nature Switzerland, 2024.
Paper Structure (7 sections, 4 figures)

This paper contains 7 sections, 4 figures.

Table of Contents

  1. Introduction
  2. Non-Euclidean geometry
  3. Excursus: Dante
  4. Geometric structures: From Ehresmann and Haefliger to Thurston
  5. Spheres and horospheres in hyperbolic space: Lobachevsky's insight
  6. Polyhedra: Andreev's theorem
  7. Volumes of polyhedra: Lobachevsky and Milnor

Figures (4)

  • Figure 1: A map of the inferno, funnel-shaped and made out of nine circles of torment; from the circle of largest to the smaller diameter, they are the realm of limbo, lust, gluttony, greed, anger, heresy, violence, and the last two circles are for fraud: flatterers and treaters. Lucifer dwells in the smallest (and deepest) circle. From the Divine Comedy Illustrated by Sandro Botticelli (1445-1510), between 1480 and 1490. Vatican Library.
  • Figure 2: Lucifer upside down. Codex Altonensis Dante, Commedia, 14th c., Bibliotheca Gimnasii Altonani, Hamburg.
  • Figure 3: Illustration of canto XXXIV of Dante's Inferno, with Lucifer upside down, and the two poets as they return to the clear world. From a manuscript in Rome, Vatican Library, Ms Urb. Lat. 365, fol. 95v. Ferrara, 1474-1482.
  • Figure 4: Illustration from Florensky's Imaginaries in geometry.