论文标题
精确形式的有界的共同体类
Bounded Cohomology Classes of Exact Forms
论文作者
论文摘要
在负弯曲的紧凑型歧管上，可以通过直接简化的集成与每个封闭形式的有界共生形式相关联。该地图的内核包含在精确形式的空间中。我们表明，在2级中，该内核很微不足道，与程度更高相比。换句话说，确切的非零$ 2 $ - 形式定义了非平凡的有限共同体学类。该结果是Barge和Ghys for Berfaces的经典定理的较高维度版本。结果，人们发现，否定弯曲的歧管的第二有界的共同体包含一个无限的维空间，其类别通过形式的整合来明确描述。这也表明，Marasco（Arxiv：2202.04419，Arxiv：2209.00560）的最新结果可以在较高的维度上应用，以获得有关某些杯子和Massey产品消失的新的非平凡结果。讨论了其他一些应用程序。
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.
