Duke Mathematical Journal

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

Nathan Broaddus

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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 proved 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.

Article information

Source
Duke Math. J. Volume 161, Number 10 (2012), 1943-1969.

Dates
First available in Project Euclid: 27 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1340801628

Digital Object Identifier
doi:10.1215/00127094-1645634

Mathematical Reviews number (MathSciNet)
MR2954621

Zentralblatt MATH identifier
1250.57032

Subjects
Primary: 57N05: Topology of $E^2$ , 2-manifolds
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

Citation

Broaddus, Nathan. Homology of the curve complex and the Steinberg module of the mapping class group. Duke Math. J. 161 (2012), no. 10, 1943--1969. doi:10.1215/00127094-1645634. https://projecteuclid.org/euclid.dmj/1340801628


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