Homomorphisms on groups of volume-preserving diffeomorphisms via fundamental groups

TL;DR

This paper explores the relationship between quasi-morphisms on the fundamental group of a closed manifold and those on the group of volume-preserving diffeomorphisms, focusing on the restriction map and flux homomorphism connection.

Contribution

It analyzes the restriction of a linear map from fundamental group quasi-morphisms to diffeomorphism group quasi-morphisms and relates it to the flux homomorphism.

Findings

The restriction map from $H^{1}(\pi_{1}(M); \\mathbb{R})$ to $H^{1}({ m Diff}_{\\Omega}^{\\infty} (M)_{0}; \\mathbb{R})$ is characterized.

The relationship between this restriction and the flux homomorphism is established.

Insights into the structure of quasi-morphisms on volume-preserving diffeomorphism 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 ${\rm Diff}_{\Omega}^{\infty} (M)_{0}$ of the group of volume-preserving diffeomorphisms of $M$. In this paper, the restriction $H^{1}(\pi_{1}(M); {\mathbb R})\to H^{1}({\rm Diff}_{\Omega}^{\infty} (M)_{0} ; {\mathbb R})$ of the linear map is studied and its relationship with the flux homomorphism is described.