Local rigidity of aspherical three-manifolds
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 393-416.

In this paper we construct, for each aspherical oriented 3-manifold M, a 2-dimensional class in the l 1 -homology of M whose norm combined with the Gromov simplicial volume of M gives a characterization of those nonzero degree maps from M to N which are homotopic to a covering map. As an application we characterize those degree one maps which are homotopic to a homeomorphism in term of isometries between the bounded cohomology groups of M and N.

Dans ce papier nous construisons, pour chaque variété de dimension trois close orientable et asphérique M, une classe d’homologie l 1 de dimension deux dans M dont la norme permet avec le volume simplicial de M de caractériser les applications de degré non-nul de M dans N qui sont homotopes à un revêtement. Comme conséquence, nous donnons un critère d’homéomorphisme pour les applications de degré un en terme d’isométries entre les groupes de cohomologie bornée de M et N.

DOI: 10.5802/aif.2708
Classification: 57M50, 51H20
Keywords: Aspherical $3$-manifolds, bounded cohomology, $l_1$-homology, non-zero degree maps, topological rigidity.
Mot clés : variétés asphériques de dimension trois, cohomologie bornée, homologie $l_1$, applications de degré non-nul, rigidité topologique.
Derbez, Pierre 1

1 LATP, UMR 6632, Centre de Mathématiques et d’Informatique, Technopole de Chateau-Gombert, 39, rue Frédéric Joliot-Curie - 13453 Marseille Cedex 13
@article{AIF_2012__62_1_393_0,
     author = {Derbez, Pierre},
     title = {Local rigidity of aspherical three-manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {393--416},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {1},
     year = {2012},
     doi = {10.5802/aif.2708},
     mrnumber = {2986274},
     zbl = {1255.57016},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2708/}
}
TY  - JOUR
AU  - Derbez, Pierre
TI  - Local rigidity of aspherical three-manifolds
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 393
EP  - 416
VL  - 62
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2708/
DO  - 10.5802/aif.2708
LA  - en
ID  - AIF_2012__62_1_393_0
ER  - 
%0 Journal Article
%A Derbez, Pierre
%T Local rigidity of aspherical three-manifolds
%J Annales de l'Institut Fourier
%D 2012
%P 393-416
%V 62
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2708/
%R 10.5802/aif.2708
%G en
%F AIF_2012__62_1_393_0
Derbez, Pierre. Local rigidity of aspherical three-manifolds. Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 393-416. doi : 10.5802/aif.2708. https://aif.centre-mersenne.org/articles/10.5802/aif.2708/

[1] Agol, I. Lower bounds on volumes of hyperbolic Haken 3-manifolds arXiv:math/9906182 (1999)

[2] Barges, J.; Ghys, E. Surfaces et cohomologie bornée, Ergodic Theory Dyn. Syst., Volume 92 (1988) no. 3, pp. 509-526 | MR | Zbl

[3] Boileau, M.; Wang, S. Non-zero degree maps and surface bundles over S 1 , J. Differential Geom., Volume 43 (1996) no. 4, pp. 789-806 | MR | Zbl

[4] Brooks, R.; Goldman, W. The Godbillon-Vey invariant of a transversely homogeneous foliation, Trans. Amer. Math. Soc., Volume 286 (1984) no. 2, pp. 651-664 | DOI | MR | Zbl

[5] Brooks, R.; Goldman, W. Volumes in Seifert space, Duke Math. J., Volume 51 (1984) no. 3, pp. 529-545 | DOI | MR | Zbl

[6] Derbez, P. Topological rigidity and Gromov simplicial volume, Comment. Math. Helv., Volume 85 (2010) no. 1, pp. 1-37 | DOI | MR

[7] Derbez, P.; Wang, S. Finiteness of mapping degrees and PSL(2,R)-volume on graph manifolds, Algebraic and Geometric Topology, Volume 9 (2009), pp. 1727-1749 | DOI | MR

[8] Fujiwara, K.; Soma, T. Bounded classes in the cohomology of manifolds, Dedicated to John Stallings on the occasion of his 65th birthday. Geom. Dedicata, Volume 92 (2002), pp. 73-85 | MR

[9] Gromov, M. Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) no. 56, pp. 5-99 | Numdam | MR | Zbl

[10] Hempel, J. 3-manifolds, Ann. of Math. Studies, 86, Princeton Univ. Press, 1976 | MR | Zbl

[11] Jaco, W.; Shalen, P.B. Seifert fibered space in 3-manifolds, Ann. of Math. Studies, 21, Mem. Amer. Math. Soc., 1979 | MR | Zbl

[12] Johannson, K. Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Mathematics, 761, Springer, Berlin, 1979 | MR | Zbl

[13] Kreck, M.; Lück, W. Topological rigidity for non-aspherical manifolds (arXiv:math.KT/0509238, to appear in Quarterly Journal of Pure and Applied mathematics)

[14] Leeb, B. 3-manifolds with(out) metrics of nonpositive curvature, Invent. Math., Volume 122 (1995) no. 2, pp. 277-289 | DOI | MR | Zbl

[15] Luecke, J.; Wu, Y. Relative Euler number and finite covers of graph manifolds, Geometric topology (Athens, GA, 1993) (AMS/IP Stud. Adv. Math, 2.1.), Amer. Math. Soc., Providence, RI, 1997, pp. 80-103 | MR | Zbl

[16] Matsumoto, S.; Morita, S. Bounded cohomology of certain groups of homeomorphisms, Proc. Amer. Math. Soc., Volume 94 (1985) no. 3, pp. 359-544 | DOI | MR | Zbl

[17] Neumann, D. A. 3-manifolds fibering over S 1 , Proc. Am. Math. Soc., Volume 58 (1979), pp. 353-356 | DOI | MR | Zbl

[18] Neumann, W. A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Am. Math. Soc., Volume 268 (1981), pp. 299-343 | DOI | MR | Zbl

[19] Rong, Y. Maps between Seifert fibered spaces of infinite π 1 , Pacific J. Math., Volume 160 (1993), pp. 143-154 | MR | Zbl

[20] Soma, T. A rigidity theorem for Haken manifolds, Math. Proc. Cambridge Philos. Soc., Volume 118 (1995) no. 1, pp. 141-160 | DOI | MR | Zbl

[21] Thurston, W. Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.), Volume 6 (1982) no. 3, pp. 357-381 | DOI | MR | Zbl

[22] Thurston, W. A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc., Volume 59 (1986) no. 339, p. i-vi and 99–130 | MR | Zbl

[23] Wang, S. The existence of maps of nonzero degree between aspherical 3-manifolds, Math. Z., Volume 208 (1991) no. 1, pp. 147-160 | DOI | MR | Zbl

[24] Wang, S. The π 1 -injectivity of self-maps of nonzero degree on 3-manifolds, Math. Ann., Volume 297 (1993) no. 1, pp. 171-189 | DOI | MR | Zbl

[25] Wang, S.; Zhou, Q. Any 3-manifold 1-dominates at most finitely many geometric 3-manifolds, Math. Ann., Volume 322 (2002) no. 3, pp. 525-535 | DOI | MR

Cited by Sources: