Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gambaudo--Ghys construction on bounded cohomology

Mitsuaki Kimura

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.

Gambaudo--Ghys construction on bounded cohomology

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.

Paper Structure

This paper contains 16 sections, 15 theorems, 48 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. (Bounded) cohomology of groups
  4. Braid groups
  5. Generalized Gambaudo--Ghys construction
  6. The braid $\gamma$
  7. The maps $\Gamma_b$ and $\Gamma$
  8. Proof of main result
  9. Construction of $\rho_\epsilon$
  10. Calculation of $\gamma(\rho_{\epsilon}(\alpha),\bar{x})$
  11. Proof of the key lemma
  12. The case of other surfaces
  13. The case of an annulus
  14. The case of a sphere
  15. The case of a torus
  16. ...and 1 more sections

Key Result

Theorem 1.1

For $n \geq 2$, the map $\overline{E\Gamma}_b \colon \overline{EH}_b^n(B_3) \to \overline{EH}_b^n(\mathcal{G})$ is injective.

Figures (5)

  • Figure 1: Open subsets in $\mathbb{D}$
  • Figure 2: Braids $\gamma(\rho_{\epsilon}(a_1),\bar{x})$ and $\beta(\bar{x}) a_1 \beta(\bar{x})^{-1}$ for $\bar{x}$ is of type (0,1,2)
  • Figure 3:
  • Figure 5: Open subsets in $\mathbb{S}$
  • Figure 6: Open subsets in $\mathbb{T}$

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1: Gromov
  • Theorem 2.2: Gromov
  • Theorem 2.3: FPS
  • Theorem 2.4: Birman
  • Lemma 3.1
  • ...and 16 more