Increasing trees and Kontsevich cycles

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 leading coefficient was computed in [Kiyoshi Igusa: Algebr. Geom. Topol. 4 (2004) 473-520]. The next coefficient was computed in [Kiyoshi Igusa: math.AT/0303157, to appear in Topology]. 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 in the last paper cited this verifies all of the formulas conjectured by Arbarello and Cornalba [J. Alg. Geom. 5 (1996) 705--749]. Mondello [math.AT/0303207, to appear in IMRN] has obtained similar results using different methods.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper26.abs.htm

Axioms for higher torsion invariants of smooth bundles

We explain the relationship between various characteristic classes for smooth manifold bundles known as ``higher torsion'' classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and transfer. We show that higher Franz-Reidemeister torsion and higher Miller-Morita-Mumford classes satisfy these axioms. Conversely, any characteristic class of smooth bundles satisfying the two axioms must be a linear combination of these two examples. We also show how higher torsion invariants can be computed using only the axioms. Finally, we explain the conjectured formula of S. Goette relating higher analytic torsion classes and higher Franz-Reidemeister torsion.Comment: 24 pages, 0 figure

Graph cohomology and Kontsevich cycles

The dual Kontsevich cycles in the double dual of associative graph homology correspond to polynomials in the Miller-Morita-Mumford classes in the integral cohomology of mapping class groups. I explain how the coefficients of these polynomials can be computed using Stasheff polyhedra and results from my previous paper GT/0207042.Comment: 36 pages, 3 figure

Maximal green sequences for cluster-tilted algebras of finite representation type

We show that, for any cluster-tilted algebra of finite representation type over an algebraically closed field, the following three definitions of a maximal green sequence are equivalent: (1) the usual definition in terms of Fomin-Zelevinsky mutation of the extended exchange matrix, (2) a forward hom-orthogonal sequence of Schurian modules, (3) the sequence of wall crossings of a generic green path. Together with [24], this completes the foundational work needed to support the author's work with P.J. Apruzzese [1], namely, to determine all lengths of all maximal green sequences for all quivers whose underlying graph is an oriented or unoriented cycle and to determine which are "linear". In an Appendix, written jointly with G. Todorov, we give a conjectural description of maximal green sequences of maximum length for any cluster-tilted algebra of finite representation type.Comment: 29 pages, revised and expanded following suggestions of two referee

Linearity of stability conditions

We study different concepts of stability for modules over a finite dimensional algebra: linear stability, given by a "central charge", and nonlinear stability given by the wall-crossing sequence of a "green path". Two other concepts, finite Harder-Narasimhan stratification of the module category and maximal forward hom-orthogonal sequences of Schurian modules, which are always equivalent to each other, are shown to be equivalent to nonlinear stability and to a maximal green sequence, defined using Fomin-Zelevinsky quiver mutation, in the case the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type $\tilde A$ and determine which are linear. The complete answer will be given in the final paper [1].Comment: 24 pages, 3 figure

The non-existence of stable Schottky forms

Let $A_g^S$ be the Satake compactification of the moduli space $A_g$ of principally polarized abelian $g$-folds and $M_g^S$ the closure of the image of the moduli space $M_g$ of genus $g$ curves in $A_g$ under the Jacobian morphism. Then $A_g^S$ lies in the boundary of $A_{g+m}^S$ for any $m$. We prove that $M_{g+m}^S$ and $A_g^S$ do not meet transversely in $A_{g+m}^S$, but rather that their intersection contains the $m$th order infinitesimal neighbourhood of $M_g^S$ in $A_g^S$. We deduce that there is no non-trivial stable Siegel modular form that vanishes on $M_g$ for every $g$. In particular, given two inequivalent positive even unimodular quadratic forms $P$ and $Q$, there is a curve whose period matrix distinguishes between the theta series of $P$ and $Q$.Comment: Corrected version, using Yamada's correct version of Fay's formula for the period matrix of a certain degenerating family of curves. To appear in Compositio Mathematic