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In this paper we construct, for each aspherical oriented -manifold , a -dimensional class in the -homology of whose norm combined with the Gromov simplicial volume of gives a characterization of those nonzero degree maps from to 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 and .
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