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.
Categories: math.GT math.GR