The Bounded Euler Class and the Symplectic Rotation Number

TL;DR

This paper extends Ghys's relationship between the bounded Euler class and the rotation number from circle homeomorphisms to the symplectic group, clarifying their connection in higher dimensions.

Contribution

It generalizes the link between the bounded Euler class and the rotation number to the symplectic group, building on prior work by Barge and Ghys.

Findings

Established a relationship between the bounded Euler class in $H_{b}^{2}(Sp(2n; eal);\mathbb{Z})$ and the symplectic rotation number.

Extended Ghys's results from circle homeomorphisms to symplectic groups.

Clarified the connection between bounded cohomology and symplectic rotation concepts.

Abstract

Ghys established the relationship between the bounded Euler class in $H_{b}^{2}(\mathrm{Homeo}_{+}(S^{1});\mathbb{Z})$ and the Poincar\'{e} rotation number, that is, he proved that the pullback of the bounded Euler class under a homomorphism $\varphi \colon \mathbb{Z} \to \mathrm{Homeo}_{+}(S^{1})$ coincides with the Poincar\'{e} rotation number of $\varphi(1)$. In this paper, we extend the above result to the symplectic group in some sense, and clarify the relationship between the bounded Euler class in $H_{b}^{2}(Sp(2n;\mathbb{R});\mathbb{Z})$ and the symplectic rotation number investigated by Barge and Ghys.