Some groups of mapping classes not realized by diffeomorphisms

TL;DR

This paper proves that certain subgroups of the mapping class group of a surface cannot be realized by diffeomorphisms, impacting the understanding of surface bundle structures and flat connections.

Contribution

It demonstrates that the subgroup of the mapping class group corresponding to the fundamental group cannot be lifted to diffeomorphisms fixing a point, providing new insights into surface bundle theory.

Findings

Subgroups of the mapping class group do not lift to diffeomorphisms.

Atiyah-Kodaira surface bundles have no invariant flat connection.

Provides an alternative proof of Morita's non-lifting theorem.

Abstract

Let S be a closed surface of genus g >= 2 and z in S a marked point. We prove that the subgroup of the mapping class group Map(S,z) corresponding to the fundamental group pi_1(S,z) of the closed surface does not lift to the group of diffeomorphisms of S fixing z. As a corollary, we show that the Atiyah-Kodaira surface bundles admit no invariant flat connection, and obtain another proof of Morita's non-lifting theorem.