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Nonzero degree maps between closed orientable three-manifolds


Author: Pierre Derbez
Journal: Trans. Amer. Math. Soc. 359 (2007), 3887-3911
MSC (2000): Primary 57M50, 51H20
DOI: https://doi.org/10.1090/S0002-9947-07-04130-X
Published electronically: March 20, 2007
MathSciNet review: 2302517
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Abstract: This paper adresses the following problem: Given a closed orientable three-manifold $ M$, are there at most finitely many closed orientable three-manifolds 1-dominated by $ M$? 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 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincaré-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold $ M$ 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of $ M$.


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  • 1. M. BOILEAU, S. WANG, Nonzero degree maps and surface bundles over $ {\S}^1$, J. Differential Geom. 43 (1996), no. 4, 789-806. MR 1412685 (98g:57023)
  • 2. E. FLAPAN, The finiteness theorem for symmetries of knots and $ 3$-manifolds with nontrivial characteristic decompositions, Special volume in honor of R. H. Bing (1914-1986). Topology Appl. 24 (1986), no. 1-3, 123-131. MR 0872482 (88d:57009)
  • 3. C. MCA. GORDON, J. LUECKE, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371-415. MR 0965210 (90a:57006a)
  • 4. M. GROMOV, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. No. 56, (1982), 5-99. MR 0686042 (84h:53053)
  • 5. C. HAYAT-LEGRAND, S.C. WANG, H. ZIESCHANG, Any $ 3$-manifold $ 1$-dominates at most finitely many $ 3$-manifolds of $ S\sp 3$-geometry, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3117-3123. MR 1908938 (2003e:55006)
  • 6. J. HEMPEL, $ 3$-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 $ 3$-manifolds, Mem. Amer. Math. Soc. 21 (1979). MR 0539411 (81c:57010)
  • 8. K. JOHANNSON, Homotopy equivalences of $ 3$-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), 35-473, Amer. Math. Soc., Providence, RI, 1997. MR 1470751
  • 10. J. LUECKE, Finite covers of $ 3$-manifolds containing essential tori, Trans. Amer. Math. Soc. 310 (1988), no. 1. 381-391. 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 $ 3$-manifolds, Amer. J. Math. 84 (1962) 1-7. MR 0142125 (25:5518)
  • 13. W. MEEKS, P. SCOTT, Finite group actions on $ 3$-manifolds, Invent. Math. 86 (1986), no. 2, 287-346. MR 0856847 (88b:57039)
  • 14. W. D. NEUMANN, D. ZAGIER, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307-332. MR 0815482 (87j:57008)
  • 15. A. REZNIKOV, Volumes of discrete groups and topological complexity of homology spheres, Math. Ann. 306 (1996), no. 3, 547-554. MR 1415078 (97i:20046)
  • 16. Y. RONG, Degree one maps between geometric $ 3$-manifolds, Trans. Amer. Math. Soc. 332 (1992), no. 1, 411-436. MR 1052909 (92j:57007)
  • 17. T. SOMA, A rigidity theorem for Haken manifolds, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 141-160. MR 1329465 (96c:57035)
  • 18. T. SOMA, Nonzero degree maps to hyperbolic $ 3$-manifolds, J. Differential Geom. 43, 517-546, 1998. MR 1669645 (2000b:57034)
  • 19. T. SOMA, Sequences of degree-one maps between geometric $ 3$-manifolds, Math. Ann. 316 (2000), no. 4, 733-742. MR 1758451 (2001b:57039)
  • 20. T. SOMA, The Gromov invariant of links, Invent. Math. 64 (1981), no. 3, 445-454. MR 0632984 (83a:57014)
  • 21. W. THURSTON, The geometry and topology of $ 3$-manifolds, Lectures Notes, Princeton Univ., 1979.
  • 22. W. THURSTON, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381. MR 0648524 (83h:57019)
  • 23. F. WALDHAUSEN, On irreducible $ 3$-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 56-88. MR 0224099 (36:7146)
  • 24. S. WANG, Q. ZHOU, Any $ 3$-manifold $ 1$-dominates at most finitely many geometric $ 3$-manifolds, Math. Ann. 322 (2002), no. 3, 525-535. MR 1895705 (2003a:57034)
  • 25. B. ZIMMERMANN, Finite group actions on Haken $ 3$-manifolds, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 499-511. 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 Aix-Marseille I, Technopole de Chateau-Gombert, 39, rue Frédéric Joliot-Curie - 13453 Marseille Cedex 13, France
Email: derbez@cmi.univ-mrs.fr

DOI: https://doi.org/10.1090/S0002-9947-07-04130-X
Keywords: Haken manifold, Seifert fibered space, geometric 3-manifold, 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.

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