Local rigidity of aspherical three-manifolds

Derbez, Pierre

Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 393-416.

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 $N$.

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 $N$.

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.

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