The Hilbert–Smith conjecture for three-manifolds

Pardon, John

doi:10.1090/S0894-0347-2013-00766-3

The Hilbert–Smith conjecture for three-manifolds

Author: John Pardon
Journal: J. Amer. Math. Soc. 26 (2013), 879-899
MSC (2010): Primary 57S10, 57M60, 20F34, 57S05, 57N10; Secondary 54H15, 55M35, 57S17
DOI: https://doi.org/10.1090/S0894-0347-2013-00766-3
Published electronically: March 19, 2013
MathSciNet review: 3037790
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Abstract: We show that every locally compact group which acts faithfully on a connected three-manifold is a Lie group. By known reductions, it suffices to show that there is no faithful action of $\mathbb Z_p$ (the $p$-adic integers) on a connected three-manifold. If $\mathbb Z_p$ acts faithfully on $M^3$, we find an interesting $\mathbb Z_p$-invariant open set $U\subseteq M$ with $H_2(U)=\mathbb Z$ and analyze the incompressible surfaces in $U$ representing a generator of $H_2(U)$. It turns out that there must be one such incompressible surface, say $F$, whose isotopy class is fixed by $\mathbb Z_p$. An analysis of the resulting homomorphism $\mathbb Z_p\to \operatorname {MCG}(F)$ gives the desired contradiction. The approach is local on $M$.

References

I. Agol\phantom{x}(mathoverflow.net/users/1345), Incompressible surfaces in an open subset of $R\,\hat {}\, 3$, MathOverflow, http://mathoverflow.net/questions/74935 (version: 2011-09-07).

J. W. Alexander, On the subdivision of 3-space by a polyhedron, Proceedings of the National Academy of Sciences of the United States of America 10 (1924), no. 1, pp. 6–8 (English).

R. H. Bing, An alternative proof that $3$-manifolds can be triangulated, Ann. of Math. (2) 69 (1959), 37–65. MR 100841, DOI https://doi.org/10.2307/1970092

Salomon Bochner and Deane Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639–653. MR 18187, DOI https://doi.org/10.2307/1969226

Yu Qing Chen, Henry H. Glover, and Craig A. Jensen, Prime order subgroups of mapping class groups, JP J. Geom. Topol. 11 (2011), no. 2, 87–99. MR 2895125

Andreas Dress, Newman’s theorems on transformation groups, Topology 8 (1969), 203–207. MR 238353, DOI https://doi.org/10.1016/0040-9383%2869%2990010-X

Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI https://doi.org/10.1007/BF02095997

A. M. Gleason, The structure of locally compact groups, Duke Math. J. 18 (1951), 85–104. MR 39730

Andrew M. Gleason, Groups without small subgroups, Ann. of Math. (2) 56 (1952), 193–212. MR 49203, DOI https://doi.org/10.2307/1969795

W. H. Gottschalk, Minimal sets: an introduction to topological dynamics, Bull. Amer. Math. Soc. 64 (1958), 336–351. MR 100048, DOI https://doi.org/10.1090/S0002-9904-1958-10223-2

Robert D. Gulliver II, Regularity of minimizing surfaces of prescribed mean curvature, Ann. of Math. (2) 97 (1973), 275–305. MR 317188, DOI https://doi.org/10.2307/1970848

A. J. S. Hamilton, The triangulation of $3$-manifolds, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 63–70. MR 407848, DOI https://doi.org/10.1093/qmath/27.1.63

William Jaco and J. Hyam Rubinstein, PL minimal surfaces in $3$-manifolds, J. Differential Geom. 27 (1988), no. 3, 493–524. MR 940116

Osamu Kakimizu, Finding disjoint incompressible spanning surfaces for a link, Hiroshima Math. J. 22 (1992), no. 2, 225–236. MR 1177053

Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI https://doi.org/10.2307/2007076

Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah; Annals of Mathematics Studies, No. 88. MR 0645390

Joo Sung Lee, Totally disconnected groups, $p$-adic groups and the Hilbert-Smith conjecture, Commun. Korean Math. Soc. 12 (1997), no. 3, 691–699. MR 1641899

Ĭozhe Maleshich, The Hilbert-Smith conjecture for Hölder actions, Uspekhi Mat. Nauk 52 (1997), no. 2(314), 173–174 (Russian); English transl., Russian Math. Surveys 52 (1997), no. 2, 407–408. MR 1480156, DOI https://doi.org/10.1070/RM1997v052n02ABEH001792

Gaven J. Martin, The Hilbert-Smith conjecture for quasiconformal actions, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 66–70. MR 1694197, DOI https://doi.org/10.1090/S1079-6762-99-00062-1

Mahan Mj, Pattern rigidity and the Hilbert-Smith conjecture, Geom. Topol. 16 (2012), no. 2, 1205–1246. MR 2946807, DOI https://doi.org/10.2140/gt.2012.16.1205

Edwin E. Moise, Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96–114. MR 48805, DOI https://doi.org/10.2307/1969769

Deane Montgomery and Leo Zippin, Small subgroups of finite-dimensional groups, Ann. of Math. (2) 56 (1952), 213–241. MR 49204, DOI https://doi.org/10.2307/1969796

Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104

M. H. A. Newman, A theorem on periodic transformations of spaces, Quart. J. Math. os-2 (1931), no. 1, 1–8.

J. Nielsen, Die struktur periodischer transformationen von flächen, Danske Vid. Selsk, Mat.-Fys. Medd. 15 (1937), 1–77.

Jakob Nielsen, Abbildungsklassen endlicher Ordnung, Acta Math. 75 (1943), 23–115 (German). MR 13306, DOI https://doi.org/10.1007/BF02404101

Robert Osserman, A survey of minimal surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. MR 0256278

Robert Osserman, A proof of the regularity everywhere of the classical solution to Plateau’s problem, Ann. of Math. (2) 91 (1970), 550–569. MR 266070, DOI https://doi.org/10.2307/1970637

C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1–26. MR 90053, DOI https://doi.org/10.2307/1970113

Piotr Przytycki and Jennifer Schultens, Contractibility of the Kakimizu complex and symmetric Seifert surfaces, Trans. Amer. Math. Soc. 364 (2012), no. 3, 1489–1508. MR 2869183, DOI https://doi.org/10.1090/S0002-9947-2011-05465-6

Frank Raymond and R. F. Williams, Examples of $p$-adic transformation groups, Ann. of Math. (2) 78 (1963), 92–106. MR 150769, DOI https://doi.org/10.2307/1970504

Dušan Repovš and Evgenij Ščepin, A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps, Math. Ann. 308 (1997), no. 2, 361–364. MR 1464908, DOI https://doi.org/10.1007/s002080050080

J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI https://doi.org/10.2307/1971131

J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 639–652. MR 654854, DOI https://doi.org/10.1090/S0002-9947-1982-0654854-8

R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI https://doi.org/10.2307/1971247

Jennifer Schultens, The Kakimizu complex is simply connected, J. Topol. 3 (2010), no. 4, 883–900. With an appendix by Michael Kapovich. MR 2746341, DOI https://doi.org/10.1112/jtopol/jtq028

P. A. Smith, Transformations of finite period. III. Newman’s theorem, Ann. of Math. (2) 42 (1941), 446–458. MR 4128, DOI https://doi.org/10.2307/1968910

Edwin H. Spanier, Cohomology theory for general spaces, Ann. of Math. (2) 49 (1948), 407–427. MR 24621, DOI https://doi.org/10.2307/1969289

Norman E. Steenrod, Universal Homology Groups, Amer. J. Math. 58 (1936), no. 4, 661–701. MR 1507191, DOI https://doi.org/10.2307/2371239

Peter Symonds, The cohomology representation of an action of $C_p$ on a surface, Trans. Amer. Math. Soc. 306 (1988), no. 1, 389–400. MR 927696, DOI https://doi.org/10.1090/S0002-9947-1988-0927696-9

Terence Tao, Hilbert’s fifth problem and related topics, manuscript, 2012, http://terrytao.wordpress.com/books/hilberts-fifth-problem-and-related-topics/.

William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI https://doi.org/10.1090/S0273-0979-1988-15685-6

Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI https://doi.org/10.2307/1970594

John J. Walsh, Light open and open mappings on manifolds. II, Trans. Amer. Math. Soc. 217 (1976), 271–284. MR 394674, DOI https://doi.org/10.1090/S0002-9947-1976-0394674-2

David Wilson, Open mappings on manifolds and a counterexample to the Whyburn conjecture, Duke Math. J. 40 (1973), 705–716. MR 320989

Hidehiko Yamabe, On the conjecture of Iwasawa and Gleason, Ann. of Math. (2) 58 (1953), 48–54. MR 54613, DOI https://doi.org/10.2307/1969819

Hidehiko Yamabe, A generalization of a theorem of Gleason, Ann. of Math. (2) 58 (1953), 351–365. MR 58607, DOI https://doi.org/10.2307/1969792

Chung-Tao Yang, $p$-adic transformation groups, Michigan Math. J. 7 (1960), 201–218. MR 120310