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 .

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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.