Nonzero degree maps between closed orientable threemanifolds
Author:
Pierre Derbez
Journal:
Trans. Amer. Math. Soc. 359 (2007), 38873911
MSC (2000):
Primary 57M50, 51H20
Published electronically:
March 20, 2007
MathSciNet review:
2302517
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Abstract: This paper adresses the following problem: Given a closed orientable threemanifold , are there at most finitely many closed orientable threemanifolds 1dominated by ? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1dominates at most finitely many orientable closed threemanifolds satisfying the PoincaréThurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable threemanifold 1dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of .
 1.
Michel
Boileau and Shicheng
Wang, Nonzero degree maps and surface bundles over
𝑆¹, J. Differential Geom. 43 (1996),
no. 4, 789–806. MR 1412685
(98g:57023)
 2.
Erica
Flapan, The finiteness theorem for symmetries of knots and
3manifolds with nontrivial characteristic decompositions, Topology
Appl. 24 (1986), no. 13, 123–131. Special
volume in honor of R. H. Bing (1914–1986). MR 872482
(88d:57009), http://dx.doi.org/10.1016/01668641(86)900532
 3.
C.
McA. Gordon and J.
Luecke, Knots are determined by their
complements, J. Amer. Math. Soc.
2 (1989), no. 2,
371–415. MR
965210 (90a:57006a), http://dx.doi.org/10.1090/S08940347198909652107
 4.
Michael
Gromov, Volume and bounded cohomology, Inst. Hautes
Études Sci. Publ. Math. 56 (1982), 5–99
(1983). MR
686042 (84h:53053)
 5.
Claude
HayatLegrand, Shicheng
Wang, and Heiner
Zieschang, Any 3manifold 1dominates at most
finitely many 3manifolds of 𝑆³geometry, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3117–3123. MR 1908938
(2003e:55006), http://dx.doi.org/10.1090/S0002993902064389
 6.
John
Hempel, 3Manifolds, Princeton University Press, Princeton, N.
J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86.
MR
0415619 (54 #3702)
 7.
William
H. Jaco and Peter
B. Shalen, Seifert fibered spaces in 3manifolds, Mem. Amer.
Math. Soc. 21 (1979), no. 220, viii+192. MR 539411
(81c:57010), http://dx.doi.org/10.1090/memo/0220
 8.
Klaus
Johannson, Homotopy equivalences of 3manifolds with
boundaries, Lecture Notes in Mathematics, vol. 761, Springer,
Berlin, 1979. MR
551744 (82c:57005)
 9.
Problems in lowdimensional topology, Geometric topology (Athens,
GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc.,
Providence, RI, 1997, pp. 35–473. MR
1470751
 10.
John
Luecke, Finite covers of 3manifolds
containing essential tori, Trans. Amer. Math.
Soc. 310 (1988), no. 1, 381–391. MR 965759
(90c:57011), http://dx.doi.org/10.1090/S00029947198809657592
 11.
Wilhelm
Magnus, Abraham
Karrass, and Donald
Solitar, Combinatorial group theory, Second revised edition,
Dover Publications, Inc., New York, 1976. Presentations of groups in terms
of generators and relations. MR 0422434
(54 #10423)
 12.
J.
Milnor, A unique decomposition theorem for 3manifolds, Amer.
J. Math. 84 (1962), 1–7. MR 0142125
(25 #5518)
 13.
William
H. Meeks III and Peter
Scott, Finite group actions on 3manifolds, Invent. Math.
86 (1986), no. 2, 287–346. MR 856847
(88b:57039), http://dx.doi.org/10.1007/BF01389073
 14.
Walter
D. Neumann and Don
Zagier, Volumes of hyperbolic threemanifolds, Topology
24 (1985), no. 3, 307–332. MR 815482
(87j:57008), http://dx.doi.org/10.1016/00409383(85)900047
 15.
Alexander
Reznikov, Volumes of discrete groups and topological complexity of
homology spheres, Math. Ann. 306 (1996), no. 3,
547–554. MR 1415078
(97i:20046), http://dx.doi.org/10.1007/BF01445265
 16.
Yong
Wu Rong, Degree one maps between geometric
3manifolds, Trans. Amer. Math. Soc.
332 (1992), no. 1,
411–436. MR 1052909
(92j:57007), http://dx.doi.org/10.1090/S00029947199210529096
 17.
Teruhiko
Soma, A rigidity theorem for Haken manifolds, Math. Proc.
Cambridge Philos. Soc. 118 (1995), no. 1,
141–160. MR 1329465
(96c:57035), http://dx.doi.org/10.1017/S0305004100073527
 18.
Teruhiko
Soma, Nonzero degree maps to hyperbolic 3manifolds, J.
Differential Geom. 49 (1998), no. 3, 517–546.
MR
1669645 (2000b:57034)
 19.
Teruhiko
Soma, Sequences of degreeone maps between geometric
3manifolds, Math. Ann. 316 (2000), no. 4,
733–742. MR 1758451
(2001b:57039), http://dx.doi.org/10.1007/s002080050352
 20.
Teruhiko
Soma, The Gromov invariant of links, Invent. Math.
64 (1981), no. 3, 445–454. MR 632984
(83a:57014), http://dx.doi.org/10.1007/BF01389276
 21.
W. THURSTON, The geometry and topology of manifolds, Lectures Notes, Princeton Univ., 1979.
 22.
William
P. Thurston, Threedimensional manifolds, Kleinian
groups and hyperbolic geometry, Bull. Amer.
Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524
(83h:57019), http://dx.doi.org/10.1090/S027309791982150030
 23.
Friedhelm
Waldhausen, On irreducible 3manifolds which are sufficiently
large, Ann. of Math. (2) 87 (1968), 56–88. MR 0224099
(36 #7146)
 24.
Shicheng
Wang and Qing
Zhou, Any 3manifold 1dominates at most finitely many geometric
3manifolds, Math. Ann. 322 (2002), no. 3,
525–535. MR 1895705
(2003a:57034), http://dx.doi.org/10.1007/s002080200003
 25.
Bruno
Zimmermann, Finite group actions on Haken 3manifolds, Quart.
J. Math. Oxford Ser. (2) 37 (1986), no. 148,
499–511. MR
868625 (88d:57029), http://dx.doi.org/10.1093/qmath/37.4.499
 1.
 M. BOILEAU, S. WANG, Nonzero degree maps and surface bundles over , J. Differential Geom. 43 (1996), no. 4, 789806. MR 1412685 (98g:57023)
 2.
 E. FLAPAN, The finiteness theorem for symmetries of knots and manifolds with nontrivial characteristic decompositions, Special volume in honor of R. H. Bing (19141986). Topology Appl. 24 (1986), no. 13, 123131. MR 0872482 (88d:57009)
 3.
 C. MCA. GORDON, J. LUECKE, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371415. MR 0965210 (90a:57006a)
 4.
 M. GROMOV, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. No. 56, (1982), 599. MR 0686042 (84h:53053)
 5.
 C. HAYATLEGRAND, S.C. WANG, H. ZIESCHANG, Any manifold dominates at most finitely many manifolds of geometry, Proc. Amer. Math. Soc. 130 (2002), no. 10, 31173123. MR 1908938 (2003e:55006)
 6.
 J. HEMPEL, manifolds, Ann. of Math. Studies, No. 86. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. xii+195 pp. MR 0415619 (54:3702)
 7.
 W. JACO, P.B. SHALEN, Seifert fibered space in manifolds, Mem. Amer. Math. Soc. 21 (1979). MR 0539411 (81c:57010)
 8.
 K. JOHANNSON, Homotopy equivalences of manifolds with boundaries, Lecture Notes in Mathematics, 761. Springer, Berlin, 1979. MR 0551744 (82c:57005)
 9.
 R. KIRBY, Problems in low dimensional topology, Edited by Rob Kirby. AMS/IP Stud. Adv. Math., 2.2, Geometric topology (Athens, GA, 1993), 35473, Amer. Math. Soc., Providence, RI, 1997. MR 1470751
 10.
 J. LUECKE, Finite covers of manifolds containing essential tori, Trans. Amer. Math. Soc. 310 (1988), no. 1. 381391. MR 0965759 (90c:57011)
 11.
 W. MAGNUS, A. KARRASS, D. SOLITAR, Combinatorial group theory, Presentations of groups in terms of generators and relations. Second revised edition. Dover Publications, Inc., New York, 1976. xii+444 pp. MR 0422434 (54:10423)
 12.
 J. MILNOR, Unique decomposition theorem for manifolds, Amer. J. Math. 84 (1962) 17. MR 0142125 (25:5518)
 13.
 W. MEEKS, P. SCOTT, Finite group actions on manifolds, Invent. Math. 86 (1986), no. 2, 287346. MR 0856847 (88b:57039)
 14.
 W. D. NEUMANN, D. ZAGIER, Volumes of hyperbolic threemanifolds, Topology 24 (1985), no. 3, 307332. MR 0815482 (87j:57008)
 15.
 A. REZNIKOV, Volumes of discrete groups and topological complexity of homology spheres, Math. Ann. 306 (1996), no. 3, 547554. MR 1415078 (97i:20046)
 16.
 Y. RONG, Degree one maps between geometric manifolds, Trans. Amer. Math. Soc. 332 (1992), no. 1, 411436. MR 1052909 (92j:57007)
 17.
 T. SOMA, A rigidity theorem for Haken manifolds, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 141160. MR 1329465 (96c:57035)
 18.
 T. SOMA, Nonzero degree maps to hyperbolic manifolds, J. Differential Geom. 43, 517546, 1998. MR 1669645 (2000b:57034)
 19.
 T. SOMA, Sequences of degreeone maps between geometric manifolds, Math. Ann. 316 (2000), no. 4, 733742. MR 1758451 (2001b:57039)
 20.
 T. SOMA, The Gromov invariant of links, Invent. Math. 64 (1981), no. 3, 445454. MR 0632984 (83a:57014)
 21.
 W. THURSTON, The geometry and topology of manifolds, Lectures Notes, Princeton Univ., 1979.
 22.
 W. THURSTON, Threedimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357381. MR 0648524 (83h:57019)
 23.
 F. WALDHAUSEN, On irreducible manifolds which are sufficiently large, Ann. of Math. 87 (1968), 5688. MR 0224099 (36:7146)
 24.
 S. WANG, Q. ZHOU, Any manifold dominates at most finitely many geometric manifolds, Math. Ann. 322 (2002), no. 3, 525535. MR 1895705 (2003a:57034)
 25.
 B. ZIMMERMANN, Finite group actions on Haken manifolds, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 499511. MR 0868625 (88d:57029)
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Additional Information
Pierre Derbez
Affiliation:
Laboratoire d’Analyse, Topologie et Probabilités, UMR 6632, Centre de Mathéma tiques et d’Informatique, Université AixMarseille I, Technopole de ChateauGombert, 39, rue Frédéric JoliotCurie  13453 Marseille Cedex 13, France
Email:
derbez@cmi.univmrs.fr
DOI:
http://dx.doi.org/10.1090/S000299470704130X
PII:
S 00029947(07)04130X
Keywords:
Haken manifold,
Seifert fibered space,
geometric 3manifold,
graph manifold,
Gromov simplicial volume,
nonzero degree maps,
Dehn filling
Received by editor(s):
March 21, 2005
Received by editor(s) in revised form:
July 18, 2005
Published electronically:
March 20, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
