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.

5 pages, to appear in Adv. Stud. Pure Math