# ANNALES DE L'INSTITUT FOURIER

Local rigidity of aspherical three-manifolds
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, p. 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 : https://doi.org/10.5802/aif.2708
Classification:  57M50,  51H20
Keywords: Aspherical $3$-manifolds, bounded cohomology, ${l}_{1}$-homology, non-zero degree maps, topological rigidity.
@article{AIF_2012__62_1_393_0,
author = {Derbez, Pierre},
title = {Local rigidity of aspherical three-manifolds},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {62},
number = {1},
year = {2012},
pages = {393-416},
doi = {10.5802/aif.2708},
zbl = {1255.57016},
mrnumber = {2986274},
language = {en},
url = {https://aif.centre-mersenne.org/item/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/item/AIF_2012__62_1_393_0/

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