Homomorphisms on groups of volume-preserving diffeomorphisms via fundamental groups

Abstract

Let $M$ be a closed manifold. Polterovich constructed a linear map from the vector space of quasi-morphisms on the fundamental group $\pi_{1}(M)$ of $M$ to the space of quasi-morphisms on the identity component $\mathrm{Diff}_{\Omega}^{\infty} (M)_{0}$ of the group of volume-preserving diffeomorphisms of $M$. In this paper, the restriction $H^{1}(\pi_{1}(M); \mathbb{R}) \to H^{1}(\mathrm{Diff}_{\Omega}^{\infty} (M)_{0}; \mathbb{R})$ of the linear map is studied and its relationship with the flux homomorphism is described.