Homology of the curve complex and 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.

23 pages, 11 figures, minor corrections

References in corpus (2)

• Stable curves and screens on fatgraphs
• Infinite topology of curve complexes and non-Poincare duality of Teichmueller modular groups

Cited by in corpus (5)

• Exhausting curve complexes by finite rigid sets
• A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)
• Cohen--Macaulay Complexes, Duality Groups, and the dualizing module of $\mathrm{Out}(F_N)$
• Problems on handlebody groups
• Finite rigid sets and homologically non-trivial spheres in the curve complex of a surface