## L. Battista - S. Francaviglia - M. Moraschini - F. Sarti - A. Savini

# Bounded Cohomology Classes of Exact Forms

created by moraschini on 09 Jan 2023

[

BibTeX]

*preprint*

**Inserted:** 9 jan 2023

**Last Updated:** 9 jan 2023

**Year:** 2022

**Abstract:**

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 have
non-trivial bounded cohomology classes. This result is the higher dimensional
version of a classical theorem by Barge and Ghys 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 (arXiv:2202.04419, arXiv:2209.00560) 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.