Bounded Cohomology Classes of Exact Forms

TL;DR

This paper investigates the relationship between bounded cohomology classes and exact forms on negatively curved manifolds, revealing that in degree 2, non-zero exact forms generate non-trivial classes, leading to an infinite-dimensional bounded cohomology space.

Contribution

It extends classical results from surfaces to higher dimensions, showing that exact 2-forms define non-trivial bounded cohomology classes and exploring applications of recent algebraic topology results.

Findings

Exact 2-forms on negatively curved manifolds define non-trivial bounded cohomology classes.

The second bounded cohomology space is infinite-dimensional.

Applications include results on cup and Massey products.

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 define 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.