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

Published electronically:
March 20, 2007

MathSciNet review:
2302517

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.**Michel Boileau and Shicheng Wang,*Non-zero degree maps and surface bundles over 𝑆¹*, J. Differential Geom.**43**(1996), no. 4, 789–806. MR**1412685****2.**Erica Flapan,*The finiteness theorem for symmetries of knots and 3-manifolds with nontrivial characteristic decompositions*, Topology Appl.**24**(1986), no. 1-3, 123–131. Special volume in honor of R. H. Bing (1914–1986). MR**872482**, 10.1016/0166-8641(86)90053-2**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**, 10.1090/S0894-0347-1989-0965210-7**4.**Michael Gromov,*Volume and bounded cohomology*, Inst. Hautes Études Sci. Publ. Math.**56**(1982), 5–99 (1983). MR**686042****5.**Claude Hayat-Legrand, Shicheng Wang, and Heiner Zieschang,*Any 3-manifold 1-dominates at most finitely many 3-manifolds of 𝑆³-geometry*, Proc. Amer. Math. Soc.**130**(2002), no. 10, 3117–3123. MR**1908938**, 10.1090/S0002-9939-02-06438-9**6.**John Hempel,*3-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619****7.**William H. Jaco and Peter B. Shalen,*Seifert fibered spaces in 3-manifolds*, Mem. Amer. Math. Soc.**21**(1979), no. 220, viii+192. MR**539411**, 10.1090/memo/0220**8.**Klaus Johannson,*Homotopy equivalences of 3-manifolds with boundaries*, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR**551744****9.**Rob Kirby (ed.),*Problems in low-dimensional 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 3-manifolds containing essential tori*, Trans. Amer. Math. Soc.**310**(1988), no. 1, 381–391. MR**965759**, 10.1090/S0002-9947-1988-0965759-2**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****12.**J. Milnor,*A unique decomposition theorem for 3-manifolds*, Amer. J. Math.**84**(1962), 1–7. MR**0142125****13.**William H. Meeks III and Peter Scott,*Finite group actions on 3-manifolds*, Invent. Math.**86**(1986), no. 2, 287–346. MR**856847**, 10.1007/BF01389073**14.**Walter D. Neumann and Don Zagier,*Volumes of hyperbolic three-manifolds*, Topology**24**(1985), no. 3, 307–332. MR**815482**, 10.1016/0040-9383(85)90004-7**15.**Alexander Reznikov,*Volumes of discrete groups and topological complexity of homology spheres*, Math. Ann.**306**(1996), no. 3, 547–554. MR**1415078**, 10.1007/BF01445265**16.**Yong Wu Rong,*Degree one maps between geometric 3-manifolds*, Trans. Amer. Math. Soc.**332**(1992), no. 1, 411–436. MR**1052909**, 10.1090/S0002-9947-1992-1052909-6**17.**Teruhiko Soma,*A rigidity theorem for Haken manifolds*, Math. Proc. Cambridge Philos. Soc.**118**(1995), no. 1, 141–160. MR**1329465**, 10.1017/S0305004100073527**18.**Teruhiko Soma,*Non-zero degree maps to hyperbolic 3-manifolds*, J. Differential Geom.**49**(1998), no. 3, 517–546. MR**1669645****19.**Teruhiko Soma,*Sequences of degree-one maps between geometric 3-manifolds*, Math. Ann.**316**(2000), no. 4, 733–742. MR**1758451**, 10.1007/s002080050352**20.**Teruhiko Soma,*The Gromov invariant of links*, Invent. Math.**64**(1981), no. 3, 445–454. MR**632984**, 10.1007/BF01389276**21.**W. THURSTON,*The geometry and topology of -manifolds*, Lectures Notes, Princeton Univ., 1979.**22.**William P. Thurston,*Three-dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 3, 357–381. MR**648524**, 10.1090/S0273-0979-1982-15003-0**23.**Friedhelm Waldhausen,*On irreducible 3-manifolds which are sufficiently large*, Ann. of Math. (2)**87**(1968), 56–88. MR**0224099****24.**Shicheng Wang and Qing Zhou,*Any 3-manifold 1-dominates at most finitely many geometric 3-manifolds*, Math. Ann.**322**(2002), no. 3, 525–535. MR**1895705**, 10.1007/s002080200003**25.**Bruno Zimmermann,*Finite group actions on Haken 3-manifolds*, Quart. J. Math. Oxford Ser. (2)**37**(1986), no. 148, 499–511. MR**868625**, 10.1093/qmath/37.4.499

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57M50,
51H20

Retrieve articles in all journals with MSC (2000): 57M50, 51H20

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.