Various exotic cohomology groups can be defined by imposing boundedness conditions on cochains. This note discusses what happens if we impose boundedness conditions in the standard deRham complex of differential forms on a Riemannian manifold and surveys relations of the resulting bounded deRham cohomology to bounded cohomology and \(l_{\infty}\)-cohomology. The problem of finding bounded primitives in all degrees and relating it to vanishing (and non vanishing) of various cohomology groups is considered. In particular, for symmetric spaces of noncompact type it is shown that in several cases bounded primitives exist.
This work consists of the following basic parts: Introduction; Bounded differential forms on symmetric spaces, bounded deRham cohomology, vanishing results for bounded deRham cohomology; Bounded cohomology and bounded differential forms, factoring through bounded differential forms; Gerstein’s \(l_{\infty}\)-cohomology; A variant of bounded valued cohomology.