2006/03/07 by Nariya Kawazumi, Kawazumi, Nariya
Mathematics · #14H15 (Primary) #20F28 #20J05 #32G15 #57M20 #57R50 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #math.AG #math.GT #msc:14H15 #msc:20F28 #msc:20J05 #msc:32G15 #msc:57M20 #msc:57R50
paper · pdf · doi:10.48550/arxiv.math/0603158
38 pages
arxiv created 2008/02/14 · arxiv updated 2009/12/01
Let \mathbb Mg, 1, g ≥ 1, be the moduli space of triples (C, P0, v) of genus g, where C is a compact Riemann surface of genus g, P0 ∈ C, and v ∈ TP0C∖\0\. Using Chen's iterated integrals we introduce a higher analogue of the period matrix for a triple (C, P0, v), \it the harmonic Magnus expansion. It induces a flat connection on a vector bundle over the space \mathbb Mg, 1, whose holonomy gives all the higher Johnson homomorphisms of the mapping class group. The connection form, which is computed as an explicit quadratic differential, induces "canonical" differential forms representing (twisted) Morita-Mumford classes and their higher relators on \mathbb Mg, 1. In particular, we construct a family of twisted differential forms on \mathbb Mg, 1 representing the (0, p+2)-twisted Morita-Mumford class m0, p+2 combinatorially parametrized by the Stasheff associahedron Kp+1.