Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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L. Battista - S. Francaviglia - M. Moraschini - F. Sarti - A. Savini

Bounded Cohomology Classes of Exact Forms

created by moraschini on 09 Jan 2023
modified by sarti on 24 May 2023


Accepted Paper

Inserted: 9 jan 2023
Last Updated: 24 may 2023

Journal: to appear in PAMS
Year: 2022

ArXiv: 2211.16125 PDF


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.

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