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

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

#### 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.

First Page:
Primary Subjects: 57N05
Secondary Subjects: 32G15
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1340801628
Digital Object Identifier: doi:10.1215/00127094-1645634
Zentralblatt MATH identifier: 06063225
Mathematical Reviews number (MathSciNet): MR2954621

### References

[ABP] J. E. Andersen, A. J. Bene, and R. C. Penner, Groupoid extensions of mapping class representations for bordered surfaces, Topology Appl. 156 (2009), 2713–2725.
Mathematical Reviews (MathSciNet): MR2556030
Zentralblatt MATH: 1196.57013
Digital Object Identifier: doi:10.1016/j.topol.2009.08.001
[As] A. Ash, Unstable cohomology of $\operatorname{SL}(n,\mathcal{O})$, J. Algebra 167 (1994), 330–342.
Mathematical Reviews (MathSciNet): MR1283290
Zentralblatt MATH: 0807.11030
Digital Object Identifier: doi:10.1006/jabr.1994.1188
[AR] A. Ash and L. Rudolph, The modular symbol and continued fractions in higher dimensions, Invent. Math. 55 (1979), 241–250.
Mathematical Reviews (MathSciNet): MR553998
Zentralblatt MATH: 0426.10023
Digital Object Identifier: doi:10.1007/BF01406842
[BF] M. Bestvina and M. Feighn, The topology at infinity of $\operatorname{Out}(F_{n})$, Invent. Math. 140 (2000), 651–692.
Mathematical Reviews (MathSciNet): MR1760754
Zentralblatt MATH: 0954.55011
Digital Object Identifier: doi:10.1007/s002220000068
[Bi] R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Math. Notes, Math. Dept., Queen Mary College, London, 1976.
Mathematical Reviews (MathSciNet): MR466344
Zentralblatt MATH: 0357.20027
[BE] R. Bieri and B. Eckmann, Groups with homological duality generalizing Poincaré duality, Invent. Math. 20 (1973), 103–124.
Mathematical Reviews (MathSciNet): MR340449
Digital Object Identifier: doi:10.1007/BF01404060
[Bir] J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213–238.
Mathematical Reviews (MathSciNet): MR243519
Zentralblatt MATH: 0167.21503
Digital Object Identifier: doi:10.1002/cpa.3160220206
[BS] A. Borel and J.-P. Serre, Corners and arithmetic groups, with appendix “Arrondissement des variétés à coins” by A. Douady and L. Hérault, Comment. Math. Helv. 48 (1973), 436–491.
Mathematical Reviews (MathSciNet): MR387495
Zentralblatt MATH: 0274.22011
Digital Object Identifier: doi:10.1007/BF02566134
[Br1] K. S. Brown, Cohomology of Groups, Grad. Texts in Math. 87, Springer, New York, 1982.
Mathematical Reviews (MathSciNet): MR672956
[Br2] K. S. Brown, Buildings, reprint of the 1989 original, Springer Monogr. Math., Springer, New York, 1998.
Mathematical Reviews (MathSciNet): MR1644630
[FM] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Math. Ser. 49, Princeton Univ. Press, Princeton, N.J., 2012.
Mathematical Reviews (MathSciNet): MR2850125
Zentralblatt MATH: 05960418
[G] P. E. Gunnells, Computing Hecke eigenvalues below the cohomological dimension, Experiment. Math. 9 (2000), 351–367.
Mathematical Reviews (MathSciNet): MR1795307
Zentralblatt MATH: 1037.11037
Digital Object Identifier: doi:10.1080/10586458.2000.10504412
Project Euclid: euclid.em/1045604670
[Ha1] J. L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121 (1985), 215–249.
Mathematical Reviews (MathSciNet): MR786348
Zentralblatt MATH: 0579.57005
Digital Object Identifier: doi:10.2307/1971172
[Ha2] J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), 157–176.
Mathematical Reviews (MathSciNet): MR830043
Zentralblatt MATH: 0592.57009
Digital Object Identifier: doi:10.1007/BF01388737
[Ha3] J. L. Harer, “The cohomology of the moduli space of curves” in Theory of Moduli (Montecatini Terme, 1985), Lecture Notes in Math. 1337, Springer, Berlin, 1988, 138–221.
Mathematical Reviews (MathSciNet): MR963064
Zentralblatt MATH: 0707.14020
[HZ] J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), 457–485.
Mathematical Reviews (MathSciNet): MR848681
Zentralblatt MATH: 0616.14017
Digital Object Identifier: doi:10.1007/BF01390325
[Hv] W. J. Harvey, “Boundary structure of the modular group” in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, N.J., 1981, 245–251.
Mathematical Reviews (MathSciNet): MR624817
Zentralblatt MATH: 0461.30036
[Ht] A. Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991), 189–194; updated version, Triangulations of surfaces, preprint, http://www.math.cornell.edu/~hatcher/Papers/TriangSurf.pdf (accessed 23 April 2012).
Mathematical Reviews (MathSciNet): MR1123262
Zentralblatt MATH: 0727.57012
Digital Object Identifier: doi:10.1016/0166-8641(91)90050-V
[HV] A. Hatcher and K. Vogtmann, The complex of free factors of a free group, Quart. J. Math. Oxford Ser. (2) 49 (1998), 459–468.
Mathematical Reviews (MathSciNet): MR1660045
Zentralblatt MATH: 0935.20015
[Iv1] N. V. Ivanov, Complexes of curves and the Teichmüller modular group (in Russian), Uspekhi Mat. Nauk 42, no. 3 (1987), 49–91, 255; English translation in Russian Math. Surveys 42 (1987), 55–107.
Mathematical Reviews (MathSciNet): MR896878
[Iv2] N. V. Ivanov, Attaching corners to Teichmüller space (in Russian), Algebra i Analiz 1, no. 5 (1989), 115–143; English translation in Leningrad Math. J. 1 (1990), 1177–1205.
Mathematical Reviews (MathSciNet): MR1036841
[Iv3] N. V. Ivanov, “Mapping class groups” in Handbook of Geometric Topology, North-Holland, Amsterdam, 2002, 523–633.
Mathematical Reviews (MathSciNet): MR1886678
[IJ] N. Ivanov and L. Ji, Infinite topology of curve complexes and non-Poincaré duality of Teichmüller modular groups, Enseign. Math. (2) 54 (2008), 381–395.
Mathematical Reviews (MathSciNet): MR2478092
[KLS] R. P. Kent, IV, C. J. Leininger, and S. Schleimer, Trees and mapping class groups, J. Reine Angew. Math. 637 (2009), 1–21.
Mathematical Reviews (MathSciNet): MR2599078
[La] S. Lang, Algebra, 3rd ed., Grad. Texts in Math. 211, Springer, New York, 2002.
Mathematical Reviews (MathSciNet): MR1878556
[Le] C. W. Lee, The associahedron and triangulations of the $n$-gon, European J. Combin. 10 (1989), 551–560.
Mathematical Reviews (MathSciNet): MR1022776
Zentralblatt MATH: 0682.52004
[LS] R. Lee and R. H. Szczarba, On the homology and cohomology of congruence subgroups, Invent. Math. 33 (1976), 15–53.
Mathematical Reviews (MathSciNet): MR422498
Zentralblatt MATH: 0332.18015
Digital Object Identifier: doi:10.1007/BF01425503
[Mo] L. Mosher, “A user’s guide to the mapping class group: Once punctured surfaces” in Geometric and Computational Perspectives on Infinite Groups (Minneapolis, Minn., and New Brunswick, N.J., 1994), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 25, Amer. Math. Soc., Providence, 1996, 101–174.
Mathematical Reviews (MathSciNet): MR1364183
Zentralblatt MATH: 0851.57015
[PM] R. C. Penner and G. McShane, Stable curves and screens on fatgraphs, preprint, arXiv:0707.1468v2 [math.GT]
arXiv: 0707.1468v2
[So] L. Solomon, “The Steinberg character of a finite group with $BN$-pair” in Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, 1969, 213–221.
Mathematical Reviews (MathSciNet): MR246951
Zentralblatt MATH: 0216.08001
[Sr] K. Strebel, Quadratic Differentials, Ergeb. Math. Grenzgeb. (3) 5, Springer, Berlin, 1984.
[Vo] K. Vogtmann, “Automorphisms of free groups and outer space” in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), Geom. Dedicata 94 (2002), 1–31.
Mathematical Reviews (MathSciNet): MR1950871
Zentralblatt MATH: 1017.20035
Digital Object Identifier: doi:10.1023/A:1020973910646