Bounded cohomology classes of exact forms

On negatively curved compact manifolds, it is possible to associate to every closed form a bounded cocycle – hence a bounded cohomology class – via integration over straight simplices. The kernel of this map is contained in the space of exact forms. We show that in degree 2 this kernel is trivial, in contrast with higher degree. In other words, exact non-zero 2-forms define non-trivial bounded cohomology classes. This result is the higher dimensional version of a classical theorem by Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] for surfaces. As a consequence, one gets that the second bounded cohomology of negatively curved manifolds contains an infinite dimensional space, whose classes are explicitly described by integration of forms. This also showcases that some recent results by Marasco [Proc. Amer. Math. Soc. 151 (2023), pp. 2707–2715] can be applied in higher dimension to obtain new non-trivial results on the vanishing of certain cup products and Massey products. Some other applications are discussed.