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Thurston’s Vision and the Virtual Fibering Theorem for 3-Manifolds

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Abstract

The vision and results of William Thurston (1946–2012) have shaped the theory of 3-dimensional manifolds for the last four decades. The high point was Perelman’s proof of Thurston’s Geometrization Conjecture which reduced 3-manifold topology for the most part to the study of hyperbolic 3-manifolds. In 1982 Thurston gave a list of 24 questions and challenges on hyperbolic 3-manifolds. The most daring one came to be known as the Virtual Fibering Conjecture. We will give some background for the conjecture and we will explain its precise content. We will then report on the recent proof of the conjecture by Ian Agol and Dani Wise.

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Notes

  1. Dehn’s Lemma says that ‘if c is an embedded curve on the boundary of a 3-manifold N such that c bounds an immersed disk in N, then it already bounds an embedded disk in ∂N’. This statement goes back to Max Dehn [12] in 1910, but Hellmuth Kneser [28, p. 260] found a gap in the proof provided by Dehn. It then took another 30 years to find a correct proof.

  2. Here recall that throughout the paper we restrict ourselves to compact manifolds; that is why \(\mathbb{R}\) is missing from our list of 1-manifolds.

  3. This topic was also discussed by Klaus Ecker [13] in an earlier Jahresbericht.

  4. In the Poincaré disk model for hyperbolic geometry the space is given by an open disk in \(\mathbb{R} ^{2}\) and the geodesics are given by the segments of Euclidean circles which intersect the given disk orthogonally. This model of hyperbolic geometry inspired M.C. Escher to create his famous woodcuts Circle Limit I, II, III and IV.

  5. The Poincaré Conjecture in dimension four was proved by Michael Freedman [15] in 1982. More precisely, he showed that any simply connected closed, topological 4-manifold is homeomorphic to S 4. It is not known, whether any simply connected, closed, differential 4-manifold is diffeomorphic to S 4. Resolving that question is often considered as the hardest problem in low-dimensional topology.

  6. Riley [46] points out that the complement of the figure-8 knot is in fact the 2-fold cover of Gieseking’s example. Hugo Gieseking was killed in France in 1915, shortly after his work on hyperbolic 3-manifold. It is conceivable that hyperbolic structures on knot complements would have been discovered much earlier if it had not been for World War I.

  7. For knots the conjecture was foreshadowed by Riley, see [46] for Riley’s account.

  8. The ICM took place in 1983 in Warsaw. It was of course supposed to take place in 1982 but it was postponed by one year because of martial law in Poland which was in effect from December 1981 to July 1983.

  9. Interestingly the phenomenon that ‘most objects are hyperbolic’ also occurs in the context of group theory. Mikhail Gromov [21] showed that in a precise sense a generic finitely presented group is ‘word hyperbolic’.

  10. It is characteristic of Ian Agol’s unassuming character that the first time he publicly mentioned this result was towards the end of an introductory lecture for graduate students in Paris. Thanks to the digital camera of the author and a blog the news of Agol’s theorem spread across the world of 3-manifold topologists within a few hours [60].

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Acknowledgements

I am very grateful to Matthias Aschenbrenner, Steve Boyer, Hermann Friedl, Hansjörg Geiges, Mark Powell, Saul Schleimer, Dan Silver and Raphael Zentner for making many helpful suggestions which greatly improved the exposition. I also wish to thank András Juhász at Keble College and Baskar Balasubramanyam at IISER Pune for providing me with the opportunity to give non-technical talks on the Virtual Fibering Theorem.

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    Stefan Friedl

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Friedl, S. Thurston’s Vision and the Virtual Fibering Theorem for 3-Manifolds. Jahresber. Dtsch. Math. Ver. 116, 223–241 (2014). https://doi.org/10.1365/s13291-014-0102-x

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