Homology of the curve complex and the Steinberg module of the mapping class group

TL;DR

This paper establishes a new explicit nontrivial homology class in the curve complex and demonstrates its role in generating the entire reduced homology under the mapping class group action, linking topology and algebraic structures.

Contribution

It provides the first explicit homologically nontrivial sphere in the curve complex and shows how its orbit generates the homology, connecting the complex's topology to the Steinberg module.

Findings

Identified the first explicit nontrivial homology sphere in the curve complex.

Proved the orbit of this sphere generates the entire reduced homology.

Connected the homology of the curve complex to the Steinberg module of the mapping class group.

Abstract

By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve complex.