On the cup product of De Rham classes in bounded cohomology

Abstract

On a negatively curved closed manifold, there exists a well-defined map $\Psi^\bullet$ associating to every closed differential form a bounded cohomology class via integration over straight simplices. Classes in the image of this map, which, a priori, depend on the fixed family of straight simplices, are usually called De Rham classes, and constitute an interesting subspace of bounded cohomology. In this paper we prove that, in sufficiently high degrees, $\Psi^\bullet$ is a homomorphism of algebras, i.e., it sends the wedge product of closed differential forms to the cup product of the associated bounded cohomology classes. The degree in which $\Psi^\bullet$ starts to preserve products depends on the boundedness of Jacobians of straight simplices. For the barycentric straightening introduced by Besson, Courtois and Gallot, this happens for degrees $\ge 3$. As a corollary, the cup product of two De Rham classes vanishes, provided that its degree exceeds the dimension of the manifold (and the degrees of both classes are $\geq 3$). This result complements vanishing results for the cup product of De Rham classes due to Marasco and to Battista et al.