Gambaudo--Ghys construction on bounded cohomology
Mitsuaki Kimura
TL;DR
This paper extends the Gambaudo--Ghys construction to bounded cohomology, proving its injectivity and demonstrating that the third bounded cohomology of certain groups of area-preserving diffeomorphisms is infinite-dimensional.
Contribution
It introduces a generalized Gambaudo--Ghys construction on bounded cohomology and establishes its injectivity, revealing new infinite-dimensional cohomology results for specific groups.
Findings
Third bounded cohomology of area-preserving diffeomorphisms on the disk is infinite-dimensional
Injectivity of the generalized Gambaudo--Ghys construction on bounded cohomology
Similar infinite-dimensional results for the sphere, torus, and annulus groups
Abstract
We consider a generalized Gambaudo--Ghys construction on bounded cohomology and prove its injectivity. As a corollary, we prove that the third bounded cohomology of the group of area-preserving diffeomorphisms on the 2-disk is infinite-dimensional. We also prove similar results for the case of the 2-sphere, the 2-torus and the annulus.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Mathematical Dynamics and Fractals