Increasing trees and Kontsevich cycles

Igusa, Kiyoshi¹ ; Kleber, Michael¹

¹ Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110, USA

Geometry & topology, Tome 8 (2004) no. 2, pp. 969-1012.

Résumé

It is known that the combinatorial classes in the cohomology of the mapping class group of punctures surfaces defined by Witten and Kontsevich are polynomials in the adjusted Miller–Morita–Mumford classes. The first two coefficients were computed by the first author in earlier papers. The present paper gives a recursive formula for all of the coefficients. The main combinatorial tool is a generating function for a new statistic on the set of increasing trees on $2n+1$ vertices. As we already explained this verifies all of the formulas conjectured by Arbarello and Cornalba. Mondello has obtained similar results using different methods.

Détail

DOI : 10.2140/gt.2004.8.969

Keywords: ribbon graphs, graph cohomology, mapping class group, Sterling numbers, hypergeometric series, Miller–Morita–Mumford classes, tautological classes

Bibliographie

[1] E Arbarello, M Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996) 705

[2] M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91

[3] J Conant, K Vogtmann, On a theorem of Kontsevich, Algebr. Geom. Topol. 3 (2003) 1167

[4] K Igusa, Combinatorial Miller–Morita–Mumford classes and Witten cycles, Algebr. Geom. Topol. 4 (2004) 473

[5] K Igusa, Higher Franz–Reidemeister torsion, 31, American Mathematical Society (2002)

[6] K Igusa, Graph cohomology and Kontsevich cycles, Topology 43 (2004) 1469

[7] K Igusa, Higher complex torsion and the framing principle, Mem. Amer. Math. Soc. 177 (2005)

[8] M Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992) 1

[9] E Y Miller, The homology of the mapping class group, J. Differential Geom. 24 (1986) 1

[10] G Mondello, Combinatorial classes on Mg,n are tautological, Int. Math. Res. Not. (2004) 2329

[11] S Morita, Characteristic classes of surface bundles, Bull. Amer. Math. Soc. (N.S.) 11 (1984) 386

[12] M Petkovšek, H S Wilf, D Zeilberger, A = B, A K Peters Ltd. (1996)

[13] R Stenli, Perechislitelnaya kombinatorika, “Mir” (1990) 440

[14] K Strebel, Quadratic differentials, , Springer (1984)