Ricci flow and diffeomorphism groups of 3-manifolds

Bamler, Richard; Kleiner, Bruce

doi:10.1090/jams/1003

Ricci flow and diffeomorphism groups of 3-manifolds
HTML articles powered by AMS MathViewer

by Richard H. Bamler and Bruce Kleiner;

J. Amer. Math. Soc. 36 (2023), 563-589

DOI: https://doi.org/10.1090/jams/1003

Published electronically: August 12, 2022

HTML | PDF | Request permission

Abstract:

We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai’s theorem for hyperbolic $3$-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except $S^3$ and $RP^3$, as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a $3$-manifold $X$, the inclusion $\operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.

References

Kouhei Asano, Homeomorphisms of prism manifolds, Yokohama Math. J. 26 (1978), no. 1, 19–25. MR 524878
Richard H. Bamler, Stability of hyperbolic manifolds with cusps under Ricci flow, Adv. Math. 263 (2014), 412–467. MR 3239144, DOI 10.1016/j.aim.2014.07.003
Richard H. Bamler, The long-time behavior of 3-dimensional Ricci flow on certain topologies, J. Reine Angew. Math. 725 (2017), 183–215. MR 3630121, DOI 10.1515/crelle-2014-0101
R. Bamler and B. Kleiner, Ricci flow and contractibility of spaces of metrics, arXiv:1909.08710, 2019.
R. Bamler and B. Kleiner, Diffeomorphism groups of prime $3$-manifolds, arXiv:2108.03302, 2021.
Richard H. Bamler and Bruce Kleiner, Uniqueness and stability of Ricci flow through singularities, Acta Math. 228 (2022), no. 1, 1–215. MR 4448680, DOI 10.4310/acta.2022.v228.n1.a1
Michel Boileau and Jean-Pierre Otal, Scindements de Heegaard et groupe des homéotopies des petites variétés de Seifert, Invent. Math. 106 (1991), no. 1, 85–107 (French). MR 1123375, DOI 10.1007/BF01243906
Francis Bonahon, Difféotopies des espaces lenticulaires, Topology 22 (1983), no. 3, 305–314 (French). MR 710104, DOI 10.1016/0040-9383(83)90016-2
Jean Cerf, Groupes d’automorphismes et groupes de difféomorphismes des variétés compactes de dimension $3$, Bull. Soc. Math. France 87 (1959), 319–329 (French). MR 116351, DOI 10.24033/bsmf.1528
Jean Cerf, La nullité de $\pi _{0}(\textrm {Diff}S^{3}$. 1. Position du problème, Séminaire Henri Cartan, 1962/63, École Norm. Sup., Paris, 1964, pp. Exp. 9-10, 27 (French). MR 198484
Jean Cerf, La nullité de $\pi _{0}(\textrm {Diff}\,S^{3}$. 2. Espaces fonctionnels liés aux décompositions d’une sphěre plongée dans R3, Séminaire Henri Cartan, 1962/63, École Norm. Sup., Paris, 1964, pp. Exp. 20, 29 (French). MR 198485
Jean Cerf, La nullité de $\pi _{0}(\textrm {Diff}S^{3}$. 3. Construction d’une section pour le revêtement ${\cal R}$, Séminaire Henri Cartan, 1962/63, École Norm. Sup., Paris, 1964, pp. Exp. 21, 25 (French). MR 198486
Jean Cerf, La nullité de $\pi _{0}(\textrm {Diff}S^{3}$. Théorèmes de fibration des espaces de plongements. Applications, Séminaire Henri Cartan, 1962/63, École Norm. Sup., Paris, 1964, pp. Exp. 8, 13 (French). MR 198483
Tobias H. Colding and William P. Minicozzi II, Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman, J. Amer. Math. Soc. 18 (2005), no. 3, 561–569. MR 2138137, DOI 10.1090/S0894-0347-05-00486-8
Sylvain E. Cappell and Julius L. Shaneson, A counterexample on the oozing problem for closed manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978) Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 627–634. MR 561242
Georges de Rham, Sur les complexes avec automorphismes, Comment. Math. Helv. 12 (1940), 191–211 (French). MR 2551, DOI 10.1007/BF01620647
Thomas Farrell, Zhou Gang, Dan Knopf, and Pedro Ontaneda, Sphere bundles with $1/4$-pinched fiberwise metrics, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6613–6630. MR 3660235, DOI 10.1090/tran/6993
Wolfgang Franz, Über die Torsion einer Überdeckung, J. Reine Angew. Math. 173 (1935), 245–254 (German). MR 1581473, DOI 10.1515/crll.1935.173.245
David Gabai, The Smale conjecture for hyperbolic 3-manifolds: $\textrm {Isom}(M^3)\simeq \textrm {Diff}(M^3)$, J. Differential Geom. 58 (2001), no. 1, 113–149. MR 1895350
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Allen Hatcher, Homeomorphisms of sufficiently large $P^{2}$-irreducible $3$-manifolds, Topology 15 (1976), no. 4, 343–347. MR 420620, DOI 10.1016/0040-9383(76)90027-6
Allen E. Hatcher, A proof of the Smale conjecture, $\textrm {Diff}(S^{3})\simeq \textrm {O}(4)$, Ann. of Math. (2) 117 (1983), no. 3, 553–607. MR 701256, DOI 10.2307/2007035
Sungbok Hong, John Kalliongis, Darryl McCullough, and J. Hyam Rubinstein, Diffeomorphisms of elliptic 3-manifolds, Lecture Notes in Mathematics, vol. 2055, Springer, Heidelberg, 2012. MR 2976322, DOI 10.1007/978-3-642-31564-0
N. V. Ivanov, Groups of diffeomorphisms of Waldhausen manifolds, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66 (1976), 172–176, 209 (Russian, with English summary). Studies in topology, II. MR 448370
N. V. Ivanov, Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 122 (1982), 72–103, 164–165, Studies in topology, IV.
N. V. Ivanov, Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 122 (1982), 72–103, 164–165 (Russian, with English summary). Studies in topology, IV. MR 661467
Bruce Kleiner and John Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587–2855. MR 2460872, DOI 10.2140/gt.2008.12.2587
Bruce Kleiner and John Lott, Singular Ricci flows I, Acta Math. 219 (2017), no. 1, 65–134. MR 3765659, DOI 10.4310/ACTA.2017.v219.n1.a4
François Laudenbach, Topologie de la dimension trois: homotopie et isotopie, Astérisque, No. 12, Société Mathématique de France, Paris, 1974 (French). With an English summary and table of contents. MR 356056
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI 10.1090/S0002-9904-1966-11484-2
G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/0303109v1, 2003.
G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245v1, 2008.
J. H. Rubinstein and J. S. Birman, One-sided Heegaard splittings and homeotopy groups of some $3$-manifolds, Proc. London Math. Soc. (3) 49 (1984), no. 3, 517–536. MR 759302, DOI 10.1112/plms/s3-49.3.517
Kurt Reidemeister, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg 11 (1935), no. 1, 102–109 (German). MR 3069647, DOI 10.1007/BF02940717
J. H. Rubinstein, On $3$-manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979), 129–137. MR 531972, DOI 10.1090/S0002-9947-1979-0531972-6
J. H. Rubinstein, Problems around 3-manifolds, Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 285–298. MR 2408251, DOI 10.2140/gtm.2007.12.285
S. Smale, Review of Cerf’s paper \cite{cerf}, 1961.
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
Rugang Ye, Ricci flow, Einstein metrics and space forms, Trans. Amer. Math. Soc. 338 (1993), no. 2, 871–896. MR 1108615, DOI 10.1090/S0002-9947-1993-1108615-3