1,002 research outputs found
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
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 be the Satake compactification of the moduli space of
principally polarized abelian -folds and the closure of the image of
the moduli space of genus curves in under the Jacobian
morphism. Then lies in the boundary of for any . We
prove that and do not meet transversely in , but
rather that their intersection contains the th order infinitesimal
neighbourhood of in . We deduce that there is no non-trivial
stable Siegel modular form that vanishes on for every . In particular,
given two inequivalent positive even unimodular quadratic forms and ,
there is a curve whose period matrix distinguishes between the theta series of
and .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
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