×

The Picard group of the moduli space of curves with level structures. (English) Zbl 1241.30015

Denoting by \(\text{Pic}({\mathcal M}_g)\) the Picard group of the moduli space \({\mathcal M}_g\) of Riemann surfaces of genus \(g\) (the set of algebraic line bundles on \({\mathcal M}_g\) under tensor products), D. Mumford [J. Anal. Math. 18, 227–244 (1967; Zbl 0173.22903)] showed that the first Chern class induces an isomorphism \(\text{Pic}({\mathcal M}_g) \cong \text{H}^2(\text{Mod}_g;\mathbb Z)\) of the Picard group with the second integral cohomology of the mapping class group \(\text{Mod}_g\) (isomorphic to \(\mathbb Z\) for large \(g\) by a result of Harer, confirming a conjecture of Mumford). The first result of the present paper is a level \(L\) version of such an isomorphism: \(\text{Pic}({\mathcal M}_g(L)) \cong \text{H}^2(\text{Mod}_g(L);\mathbb Z)\) for each integer \(L \geq 2\) not divisible by 4 (which is assumed for purely technical reasons); here \(\text{Mod}_g(L)\) denotes the level \(L\) subgroup of \(\text{Mod}_g\), i.e. all mapping classes acting trivially on the first homology of the surface with coefficients in the integers mod \(L\), and \({\mathcal M}_g(L)\) the corresponding finite covering of \({\mathcal M}_g(L)\) (the moduli space of curves with level \(L\) structures; as the author notes, these are fine moduli spaces rather than merely coarse ones, and a key point here is the fact that Riemann surfaces can have automorphisms whereas Riemann surfaces with fixed level structures cannot). The bulk of the present paper is then devoted to the following.
“We determine the divisibility properties of the standard line bundles over these moduli spaces, and we calculate the second integral cohomology group of the level \(L\) subgroup of the mapping class group. (In a previous paper [Adv. Math. 229, No. 2, 1205–1234 (2012; Zbl 1250.14019)] the author determined this rationally.) This entails calculating the abelianization of the level \(L\) subgroup of the mapping class group, generalizing previous results of Perron, Sato and the author. Finally, along the way we calculate the first homology of the symplectic group \(\text{Sp}_{2g}(\mathbb Z/L)\) with coefficients in the adjoint representation.”

MSC:

30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57M50 General geometric structures on low-dimensional manifolds
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] E. Arbarello and M. Cornalba, The Picard groups of the moduli spaces of curves , Topology 26 (1987), 153-171. · Zbl 0625.14014
[2] W. L. Baily, Jr., On the moduli of Jacobian varieties , Ann. of Math. (2) 71 (1960), 303-314. · Zbl 0178.55001
[3] H. Bass, Algebraic K-theory , W. A. Benjamin, New York, 1968.
[4] B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s Property (T) , New Math. Monogr. 11 , Cambridge Univ. Press, Cambridge, 2008. · Zbl 1146.22009
[5] J. S. Birman, On Siegel’s modular group , Math. Ann. 191 (1971), 59-68. · Zbl 0208.10601
[6] J. S. Birman and R. Craggs, The \mu -invariant of 3 -manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2 -manifold , Trans. Amer. Math. Soc. 237 (1978), 283-309. · Zbl 0383.57006
[7] A. Borel, “Properties and linear representations of Chevalley groups” in Seminar on Algebraic Groups and Related Finite Groups (Princeton, N. J., 1968/69) , Lecture Notes in Math. 131 , Springer, Berlin, 1970, 1-55. · Zbl 0197.30501
[8] A. Borel, Stable real cohomology of arithmetic groups , Ann. Sci. École Norm. Sup. (4) 7 (1974), 235-272 (1975). · Zbl 0316.57026
[9] A. Borel, “Stable real cohomology of arithmetic groups, II” in Manifolds and Lie Groups (Notre Dame, Ind., 1980) , Progr. Math. 14 , Birkhäuser, Boston, 1981, 21-55. · Zbl 0483.57026
[10] N. Broaddus, B. Farb, and A. Putman, Irreducible Sp -representations and subgroup distortion in the mapping class group , Comment. Math. Helv. 86 (2011), 537-556. · Zbl 1295.57021
[11] K. S. Brown, Cohomology of Groups , corrected reprint of the 1982 original, Grad. Texts in Math. 87 , Springer, New York, 1994.
[12] R. Charney, A generalization of a theorem of Vogtmann , J. Pure Appl. Algebra 44 (1987), 107-125. · Zbl 0615.20024
[13] P. E. Conner, Lectures on the Action of a Finite Group , Lecture Notes in Math. 73 , Springer, Berlin, 1968. · Zbl 0177.26302
[14] C. W. Curtis, On projective representations of certain finite groups , Proc. Amer. Math. Soc. 11 (1960), 852-860. · Zbl 0098.25303
[15] P. Deligne, Extensions centrales non résiduellement finies de groupes arithmétiques , C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 4, A203-A208. · Zbl 0416.20042
[16] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus , Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75-109. · Zbl 0181.48803
[17] R. K. Dennis and M. R. Stein, “The functor K 2 : A survey of computations and problems” in Algebraic K-theory, II: “Classical” Algebraic K-theory and Connections with Arithmetic (Battelle Memorial Inst., Seattle, Wash., 1972) , Lecture Notes in Math. 342 , Springer, Berlin, 1973, 243-280. · Zbl 0271.18011
[18] E. B. Dynkin, The structure of semi-simple algebras , Amer. Math. Soc. Translation 17 (1950), 143 pp. · Zbl 0052.26202
[19] B. Farb, “Some problems on mapping class groups and moduli space” in Problems on Mapping Class Groups and Related Topics , Proc. Sympos. Pure Math. 74 , Amer. Math. Soc., Providence, 2006, 11-55. · Zbl 1191.57015
[20] B. Farb and D. Margalit, A Primer on Mapping Class Groups , Princeton Math. Ser., Princeton Univ. Press, Princeton, N.J., 2012. · Zbl 1245.57002
[21] G. Farkas, The birational type of the moduli space of even spin curves , Adv. Math. 223 (2010), 433-443. · Zbl 1183.14020
[22] E. Freitag, Singular Modular Forms and Theta Relations , Lecture Notes in Math. 1487 , Springer, Berlin, 1991. · Zbl 0744.11024
[23] H. Grauert and R. Remmert, Komplexe Räume , Math. Ann. 136 (1958), 245-318. · Zbl 0087.29003
[24] A. Grothendieck and M. Raynaud, Revêtements étales et groupe fondamental , Séminaire de Géométrie Algébrique du Bois-Marie 1960-61 (SGA 1), Doc. Math. (Paris) 3 , Soc. Math. France, Paris, 2003.
[25] R. Hain, “Torelli groups and geometry of moduli spaces of curves” in Current Topics in Complex Algebraic Geometry (Berkeley, Calif., 1992/93) , Math. Sci. Res. Inst. Publ. 28 , Cambridge Univ. Press, Cambridge, 1995, 97-143. · Zbl 0868.14006
[26] R. Hain, “Moduli of Riemann surfaces, transcendental aspects” in School on Algebraic Geometry (Trieste, 1999) , ICTP Lect. Notes 1 , Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000, 293-353. · Zbl 0995.14007
[27] J. Harer, The second homology group of the mapping class group of an orientable surface , Invent. Math. 72 (1983), 221-239. · Zbl 0533.57003
[28] J. Harris and I. Morrison, Moduli of Curves , Springer, New York, 1998. · Zbl 0913.14005
[29] D. F. Holt, B. Eick and E. A. O’Brien, Handbook of Computational Group Theory , Chapman and Hall/CRC, Boca Raton, Fla., 2005. · Zbl 1091.20001
[30] J. Huebschmann, Group extensions, crossed pairs and an eight term exact sequence , J. Reine Angew. Math. 321 (1981), 150-172. · Zbl 0441.20033
[31] J. Igusa, On the graded ring of theta-constants , Amer. J. Math. 86 (1964), 219-246. · Zbl 0146.31703
[32] J. Igusa, Theta Functions , Springer, New York, 1972. · Zbl 0251.14016
[33] S. Illman, Existence and uniqueness of equivariant triangulations of smooth proper G-manifolds with some applications to equivariant Whitehead torsion , J. Reine Angew. Math. 524 (2000), 129-183. · Zbl 0953.57014
[34] N. Jacobson, Classes of restricted Lie algebras of characteristic p, I , Amer. J. Math. 63 (1941), 481-515. · Zbl 0025.30302
[35] D. Johnson, Homeomorphisms of a surface which act trivially on homology , Proc. Amer. Math. Soc. 75 (1979), 119-125. · Zbl 0407.57003
[36] D. Johnson, An abelian quotient of the mapping class group \(\mathcal{I}_{g}\) , Math. Ann. 249 (1980), 225-242. · Zbl 0409.57009
[37] D. Johnson, Quadratic forms and the Birman-Craggs homomorphisms , Trans. Amer. Math. Soc. 261 (1980), 235-254. · Zbl 0457.57006
[38] D. Johnson, “A survey of the Torelli group” in Low-dimensional Topology (San Francisco, Calif., 1981) , Contemp. Math. 20 , Amer. Math. Soc., Providence, 1983, 165-179. · Zbl 0553.57002
[39] D. Johnson, The structure of the Torelli group, I: A finite set of generators for \(\mathcal{I}\) , Ann. of Math. (2) 118 (1983), 423-442. · Zbl 0549.57006
[40] D. Johnson, The structure of the Torelli group, III: The abelianization of \(\mathcal{T}\) , Topology 24 (1985), 127-144. · Zbl 0571.57010
[41] D. A. Kazhdan [Každan], On the connection of the dual space of a group with the structure of its closed subgroups , Funkcional. Anal. i Priložen. 1 (1967), 71-74. · Zbl 0168.27602
[42] S. Kobayashi, Differential Geometry of Complex Vector Bundles , Publ. Math. Soc. Japan 15 , Princeton Univ. Press, Princeton, N.J., 1987. · Zbl 0708.53002
[43] J. Kock and I. Vainsencher, An Invitation to Quantum Cohomology , Progr. Math. 249 , Birkhäuser, Boston, 2007. · Zbl 1114.14001
[44] M. Korkmaz, Low-dimensional homology groups of mapping class groups: A survey , Turkish J. Math. 26 (2002), 101-114. · Zbl 1026.57015
[45] R. K. Lashof, J. P. May and G. B. Segal, “Equivariant bundles with abelian structural group” in Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) , Contemp. Math. 19 , Amer. Math. Soc., Providence, 1983, 167-176. · Zbl 0526.55020
[46] R. Lee and R. H. Szczarba, On the homology and cohomology of congruence subgroups , Invent. Math. 33 (1976), 15-53. · Zbl 0332.18015
[47] S. Lojasiewicz, Triangulation of semi-analytic sets , Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 449-474. · Zbl 0128.17101
[48] S. Mac Lane, Homology , reprint of the 1975 edition, Springer, Berlin, 1995.
[49] J. D. McCarthy, On the first cohomology group of cofinite subgroups in surface mapping class groups , Topology 40 (2001), 401-418. · Zbl 0976.14023
[50] D. McCullough and A. Miller, The genus 2 Torelli group is not finitely generated , Topology Appl. 22 (1986), 43-49. · Zbl 0579.57007
[51] J. Milnor, Introduction to Algebraic K-theory , Ann. of Math. Stud. 72 , Princeton Univ. Press, Princeton, N.J., 1971. · Zbl 0237.18005
[52] B. Mirzaii and W. van der Kallen, Homology stability for unitary groups , Doc. Math. 7 (2002), 143-166 (electronic). · Zbl 0999.19005
[53] D. Mumford, Abelian quotients of the Teichmüller modular group , J. Analyse Math. 18 (1967), 227-244. · Zbl 0173.22903
[54] D. Mumford, Abelian Varieties , corrected reprint of the 2nd (1974) edition, Tata Inst. Fund. Res. Studies in Math. 5 , Tata Inst. Fund. Res., Bombay, 2008.
[55] D. Mumford, Tata Lectures on Theta, I , Birkhäuser, Boston, Mass., 1983.
[56] M. Newman and J. R. Smart, Symplectic modulary groups , Acta Arith. 9 (1964), 83-89. · Zbl 0135.06502
[57] B. Perron, Filtration de Johnson et groupe de Torelli modulo p, p premier , C. R. Math. Acad. Sci. Paris 346 (2008), 667-670. · Zbl 1147.57020
[58] W. Pitsch, Un calcul élémentaire de \(H_{2}(\mathcal{M}_{g,1},\mathbf{Z})\) pour g \geq 4, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 667-670. · Zbl 0934.57016
[59] H. Pollatsek, First cohomology groups of some linear groups over fields of characteristic two , Illinois J. Math. 15 (1971), 393-417. · Zbl 0218.20039
[60] J. Powell, Two theorems on the mapping class group of a surface , Proc. Amer. Math. Soc. 68 (1978), 347-350. · Zbl 0391.57009
[61] A. Putman, An infinite presentation of the Torelli group , Geom. Funct. Anal. 19 (2009), 591-643. · Zbl 1178.57001
[62] A. Putman, The abelianization of the level L mapping class group , preprint, [math.GT] 0803.0539v2
[63] A. Putman, The second rational homology group of the moduli space of curves with level structures , Adv. Math. 229 (2012), 1205-1234. · Zbl 1250.14019
[64] M. Sato, The abelianization of the level d mapping class group , J. Topol. 3 (2010), 847-882. · Zbl 1209.57012
[65] G. B. Seligman, Modular Lie Algebras , Springer, New York, 1967. · Zbl 0189.03201
[66] M. R. Stein, Surjective stability in dimension 0 for K 2 and related functors , Trans. Amer. Math. Soc. 178 (1973), 165-191. · Zbl 0267.18015
[67] R. Steinberg, “Générateurs, relations et revetements de groupes algébriques” in Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) , Librairie Universitaire, Louvain, 1962, 113-127. · Zbl 0272.20036
[68] W. Thurston, The geometry and topology of three-manifolds , notes, Princeton University, Princeton, N.J., 1980, .
[69] H. Völklein, The 1-cohomology of the adjoint module of a Chevalley group , Forum Math. 1 (1989), 1-13. · Zbl 0649.20042
[70] R. O. Wells, Jr., Differential Analysis on Complex Manifolds , 3rd. ed., Grad. Texts in Math. 65 , Springer, New York, 2008.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.