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 , are there at most finitely many closed orientable three-manifolds 1-dominated 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 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 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of .

**1.**M. BOILEAU, S. WANG,*Nonzero degree maps and surface bundles over*, J. Differential Geom. 43 (1996), no. 4, 789-806. 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 (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 -manifold -dominates at most finitely many -manifolds of -geometry*, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3117-3123. 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), 35-473, 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. 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 -manifolds*, Amer. J. Math. 84 (1962) 1-7. MR**0142125 (25:5518)****13.**W. MEEKS, P. SCOTT,*Finite group actions on -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 -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 -manifolds*, J. Differential Geom. 43, 517-546, 1998. MR**1669645 (2000b:57034)****19.**T. SOMA,*Sequences of degree-one maps between geometric -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 -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 -manifolds which are sufficiently large*, Ann. of Math. 87 (1968), 56-88. MR**0224099 (36:7146)****24.**S. WANG, Q. ZHOU,*Any -manifold -dominates at most finitely many geometric -manifolds*, Math. Ann. 322 (2002), no. 3, 525-535. MR**1895705 (2003a:57034)****25.**B. ZIMMERMANN,*Finite group actions on Haken -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.