A geometric description of the homology of surface bundles

TL;DR

This paper provides a geometric framework for understanding the homology of surface bundles over surfaces using the holonomy map, addressing cases with closed or boundary surfaces and clarifying previous inaccuracies.

Contribution

It introduces a geometric description of homology classes in surface bundles and corrects earlier theoretical errors with new insights.

Findings

Homology classes can be described geometrically via the holonomy map.

The paper clarifies the homology structure for bundles over surfaces with boundary.

It explains why previous strategies failed and offers corrected observations.

Abstract

In the present note we describe geometrically the homology classes in the total space of a surface bundle over a surface in terms of the holonomy map. We treat the cases where the base surface is closed or has one boundary component. We replace the wrong theorem of the previous version by some observations on the homology of surface bundles and explain why our previous strategy failed.