×

zbMATH — the first resource for mathematics

The high-dimensional cohomology of the moduli space of curves with level structures. (English) Zbl 07198108
Summary: We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the moduli space of curves is at least \(g - 2\). Well known conjectures of Looijenga would imply that this is sharp.

MSC:
14D22 Fine and coarse moduli spaces
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
Software:
MathOverflow
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andrews, G. E.: The Theory of Partitions. Addison-Wesley, Reading, MA (1976) Zbl 0371.10001 MR 0557013 · Zbl 0371.10001
[2] Broaddus, N.: Homology of the curve complex and the Steinberg module of the mapping class group, Duke Math. J.161, 1943-1969 (2012)Zbl 1250.57032 MR 2954621 · Zbl 1250.57032
[3] Brown, K. S.: Buildings. Springer, New York (1989)Zbl 0715.20017 MR 0969123
[4] Chaudhuri, C.: The cohomological excess of certain moduli spaces of curves of genusg. Int. Math. Res. Notices2015, 1056-1074Zbl 1373.14027 MR 3340347 · Zbl 1373.14027
[5] Church, T., Farb, B., Putman, A.: A stability conjecture for the unstable cohomology of SLn(Z), mapping class groups, and Aut(Fn). In: Algebraic Topology: Applications and New Directions, Contemp. Math. 620, Amer. Math. Soc., Providence, RI, 55-70 (2014)Zbl 1377.11065 MR 3290086 · Zbl 1377.11065
[6] Faber, C., Looijenga, E.: Remarks on moduli of curves. In: Moduli of Curves and Abelian Varieties, Aspects Math. E33, Vieweg, Braunschweig, 23-45 (1999) Zbl 0978.14028 MR 1722537 · Zbl 0978.14028
[7] Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Math. Ser. 49, Princeton Univ. Press, Princeton, NJ (2012)Zbl 1245.57002 MR 2850125 · Zbl 1245.57002
[8] Fontanari, C., Pascolutti, S.: An affine open covering of Mgforg≤5. Geom. Dedicata158, 61-68 (2012)Zbl 1253.14028 MR 2922703 · Zbl 1253.14028
[9] Grove, L. C.: Classical Groups and Geometric Algebra. Grad. Stud. Math. 39, Amer. Math. Soc., Providence, RI (2002)Zbl 0990.20001 MR 1859189
[10] Hain, R. M.: Torelli groups and geometry of moduli spaces of curves. In: Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ. 28, Cambridge Univ. Press, Cambridge, 97-143 (1995)Zbl 0868.14006 MR 1397061 · Zbl 0868.14006
[11] Harer, J. L.: Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math. (2)121, 215-249 (1985)Zbl 0579.57005 MR 0786348 · Zbl 0579.57005
[12] Harer, J. L.: The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math.84, 157-176 (1986)Zbl 0592.57009 MR 0830043 · Zbl 0592.57009
[13] Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math.85, 457-485 (1986)Zbl 0616.14017 MR 0848681 · Zbl 0616.14017
[14] Hartshorne, R.: Cohomological dimension of algebraic varieties. Ann. of Math. (2) 88, 403-450 (1968)Zbl 0169.23302 MR 0232780 · Zbl 0169.23302
[15] Hatcher, A.: On triangulations of surfaces. Topology Appl.40, 189-194 (1991) Zbl 0727.57012 MR 1123262 · Zbl 0727.57012
[16] Hatcher, A., Vogtmann, K.: Tethers and homology stability for surfaces. Algebr. Geom. Topol.17, 1871-1916 (2017)Zbl 06762603 MR 3677942 · Zbl 1439.20063
[17] Kent, R. P., IV, Leininger, C. J., Schleimer, S.: Trees and mapping class groups. J. Reine Angew. Math.637, 1-21 (2009)Zbl 1190.57014 MR 2599078 · Zbl 1190.57014
[18] Lee, R., Szczarba, R. H.: On the homology and cohomology of congruence subgroups. Invent. Math.33, 15-53 (1976)Zbl 0332.18015 MR 0422498 · Zbl 0332.18015
[19] Lucia (http://mathoverflow.net/users/38624/lucia): Identity involving a sum over all partitions ofn.http://mathoverflow.net/q/250862(2016)
[20] Madsen, I., Weiss, M.: The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math. (2)165, 843-941 (2007)Zbl 1156.14021 MR 2335797 · Zbl 1156.14021
[21] Mondello, G.: On the cohomological dimension of the moduli space of Riemann surfaces. Duke Math. J.166, 1463-1515 (2017)Zbl 1388.32007 MR 3659940 · Zbl 1388.32007
[22] Morita, S., Sakasai, T., Suzuki, M.: Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra. Duke Math. J.162, 965-1002 (2013)Zbl 1308.17021 MR 3047471 · Zbl 1308.17021
[23] Putman, A., The second rational homology group of the moduli space of curves with level structures, Adv. Math.229, 1205-1234 (2012)Zbl 1250.14019 MR 2855091 · Zbl 1250.14019
[24] Serre, J.-P.: Sur la cohomologie des vari´et´es alg´ebriques. J. Math. Pures Appl. (9)36, 1-16 (1957)Zbl 0078.34604 MR 0083813
[25] Solomon, L.: The Steinberg character of a finite group withBN-pair. In: Theory of Finite Groups (Cambridge, MA, 1968), Benjamin, New York, 213-221 (1969) Zbl 0216.08001 MR 0246951
[26] Stanley, R.
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.