The Hilbert-Smith conjecture for three-manifolds. (English) Zbl 1273.57024

Pardon, John

J. Am. Math. Soc. 26, No. 3, 879-899 (2013).

The author affirms the Hilbert–Smith conjecture for manifolds of dimension \(3\). He shows that there is no faithful action of \(\mathbb Z_p\) (the \(p\)–adic integers) on any connected three–manifold. Equivalently, if a locally compact topological group \(G\) acts faithfully on some connected \(3\)–manifold, then \(G\) is a Lie group.

Reviewer: Karl Heinz Dovermann (Honolulu)

MSC:

57S10	Compact groups of homeomorphisms
57M60	Group actions on manifolds and cell complexes in low dimensions
20F34	Fundamental groups and their automorphisms (group-theoretic aspects)
57S05	Topological properties of groups of homeomorphisms or diffeomorphisms
57N10	Topology of general \(3\)-manifolds (MSC2010)
54H15	Transformation groups and semigroups (topological aspects)
55M35	Finite groups of transformations in algebraic topology (including Smith theory)
57S17	Finite transformation groups

Keywords:

Hilbert-Smith Conjecture

Software:

MathOverflow

References:

[1]	I. Agol(mathoverflow.net/users/1345), Incompressible surfaces in an open subset of \( R\,\hat {}\, 3\), MathOverflow, http://mathoverflow.net/questions/74935 (version: 2011-09-07).
[2]	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). · JFM 50.0659.01
[3]	R. H. Bing, An alternative proof that 3-manifolds can be triangulated, Ann. of Math. (2) 69 (1959), 37 – 65. · Zbl 0106.16604 · doi:10.2307/1970092
[4]	Salomon Bochner and Deane Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639 – 653. · Zbl 0061.04407 · doi:10.2307/1969226
[5]	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. · Zbl 1230.57002
[6]	Andreas Dress, Newman’s theorems on transformation groups, Topology 8 (1969), 203 – 207. · Zbl 0176.53201 · doi:10.1016/0040-9383(69)90010-X
[7]	Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in 3-manifolds, Invent. Math. 71 (1983), no. 3, 609 – 642. · Zbl 0482.53045 · doi:10.1007/BF02095997
[8]	A. M. Gleason, The structure of locally compact groups, Duke Math. J. 18 (1951), 85 – 104. · Zbl 0044.01901
[9]	Andrew M. Gleason, Groups without small subgroups, Ann. of Math. (2) 56 (1952), 193 – 212. · Zbl 0049.30105 · doi:10.2307/1969795
[10]	W. H. Gottschalk, Minimal sets: an introduction to topological dynamics, Bull. Amer. Math. Soc. 64 (1958), 336 – 351. · Zbl 0085.17401
[11]	Robert D. Gulliver II, Regularity of minimizing surfaces of prescribed mean curvature, Ann. of Math. (2) 97 (1973), 275 – 305. · Zbl 0246.53053 · doi:10.2307/1970848
[12]	A. J. S. Hamilton, The triangulation of 3-manifolds, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 63 – 70. · Zbl 0318.57003 · doi:10.1093/qmath/27.1.63
[13]	William Jaco and J. Hyam Rubinstein, PL minimal surfaces in 3-manifolds, J. Differential Geom. 27 (1988), no. 3, 493 – 524. · Zbl 0652.57005
[14]	Osamu Kakimizu, Finding disjoint incompressible spanning surfaces for a link, Hiroshima Math. J. 22 (1992), no. 2, 225 – 236. · Zbl 0774.57006
[15]	Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235 – 265. · Zbl 0528.57008 · doi:10.2307/2007076
[16]	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. · Zbl 0361.57004
[17]	Joo Sung Lee, Totally disconnected groups, \?-adic groups and the Hilbert-Smith conjecture, Commun. Korean Math. Soc. 12 (1997), no. 3, 691 – 699. · Zbl 0944.54029
[18]	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. · Zbl 0988.57021 · doi:10.1070/RM1997v052n02ABEH001792
[19]	Gaven J. Martin, The Hilbert-Smith conjecture for quasiconformal actions, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 66 – 70. · Zbl 0979.30013
[20]	Mahan Mj, Pattern rigidity and the Hilbert-Smith conjecture, Geom. Topol. 16 (2012), no. 2, 1205 – 1246. · Zbl 1259.20054 · doi:10.2140/gt.2012.16.1205
[21]	Edwin E. Moise, Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96 – 114. · Zbl 0048.17102 · doi:10.2307/1969769
[22]	Deane Montgomery and Leo Zippin, Small subgroups of finite-dimensional groups, Ann. of Math. (2) 56 (1952), 213 – 241. · Zbl 0049.30106 · doi:10.2307/1969796
[23]	Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. · Zbl 0068.01904
[24]	M. H. A. Newman, A theorem on periodic transformations of spaces, Quart. J. Math. os-2 (1931), no. 1, 1-8. · Zbl 0001.22703
[25]	J. Nielsen, Die struktur periodischer transformationen von flächen, Danske Vid. Selsk, Mat.-Fys. Medd. 15 (1937), 1-77. · JFM 63.0553.03
[26]	Jakob Nielsen, Abbildungsklassen endlicher Ordnung, Acta Math. 75 (1943), 23 – 115 (German). · Zbl 0027.26601 · doi:10.1007/BF02404101
[27]	Robert Osserman, A survey of minimal surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. · Zbl 0209.52901
[28]	Robert Osserman, A proof of the regularity everywhere of the classical solution to Plateau’s problem, Ann. of Math. (2) 91 (1970), 550 – 569. · Zbl 0194.22302 · doi:10.2307/1970637
[29]	C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1 – 26. · Zbl 0078.16402 · doi:10.2307/1970113
[30]	Piotr Przytycki and Jennifer Schultens, Contractibility of the Kakimizu complex and symmetric Seifert surfaces, Trans. Amer. Math. Soc. 364 (2012), no. 3, 1489 – 1508. · Zbl 1239.57025
[31]	Frank Raymond and R. F. Williams, Examples of \?-adic transformation groups, Ann. of Math. (2) 78 (1963), 92 – 106. · Zbl 0178.26003 · doi:10.2307/1970504
[32]	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. · Zbl 0879.57025 · doi:10.1007/s002080050080
[33]	J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1 – 24. · Zbl 0462.58014 · doi:10.2307/1971131
[34]	J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 639 – 652. · Zbl 0527.58008
[35]	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. · Zbl 0431.53051 · doi:10.2307/1971247
[36]	Jennifer Schultens, The Kakimizu complex is simply connected, J. Topol. 3 (2010), no. 4, 883 – 900. With an appendix by Michael Kapovich. · Zbl 1222.57006 · doi:10.1112/jtopol/jtq028
[37]	P. A. Smith, Transformations of finite period. III. Newman’s theorem, Ann. of Math. (2) 42 (1941), 446 – 458. · JFM 67.0743.01 · doi:10.2307/1968910
[38]	Edwin H. Spanier, Cohomology theory for general spaces, Ann. of Math. (2) 49 (1948), 407 – 427. · Zbl 0035.24801 · doi:10.2307/1969289
[39]	Norman E. Steenrod, Universal Homology Groups, Amer. J. Math. 58 (1936), no. 4, 661 – 701. · Zbl 0015.17901 · doi:10.2307/2371239
[40]	Peter Symonds, The cohomology representation of an action of \?_{\?} on a surface, Trans. Amer. Math. Soc. 306 (1988), no. 1, 389 – 400. · Zbl 0645.57008
[41]	Terence Tao, Hilbert’s fifth problem and related topics, manuscript, 2012, http://terrytao.wordpress.com/books/hilberts-fifth-problem-and-related-topics/. · Zbl 1298.22001
[42]	William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417 – 431. · Zbl 0674.57008
[43]	Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56 – 88. · Zbl 0157.30603 · doi:10.2307/1970594
[44]	John J. Walsh, Light open and open mappings on manifolds. II, Trans. Amer. Math. Soc. 217 (1976), 271 – 284. · Zbl 0321.57011
[45]	David Wilson, Open mappings on manifolds and a counterexample to the Whyburn conjecture, Duke Math. J. 40 (1973), 705 – 716. · Zbl 0273.54008
[46]	Hidehiko Yamabe, On the conjecture of Iwasawa and Gleason, Ann. of Math. (2) 58 (1953), 48 – 54. · Zbl 0053.01601 · doi:10.2307/1969819
[47]	Hidehiko Yamabe, A generalization of a theorem of Gleason, Ann. of Math. (2) 58 (1953), 351 – 365. · Zbl 0053.01602 · doi:10.2307/1969792
[48]	Chung-Tao Yang, \?-adic transformation groups, Michigan Math. J. 7 (1960), 201 – 218. · Zbl 0094.17502