Harmonic Magnus Expansion on the Universal Family of Riemann Surfaces

Abstract

Let ${\mathbb{M}}_{g,1}$, $g\ge 1$, be the moduli space of triples $(C,P_0,v)$ of genus $g$, where $C$ is a compact Riemann surface of genus $g$, $P_0\in C$, and $v\in T_{P_0}C\setminus \{0\}$. Using Chen's iterated integrals we introduce a higher analogue of the period matrix for a triple $(C,P_0,v)$, {\it the harmonic Magnus expansion}. It induces a flat connection on a vector bundle over the space ${\mathbb{M}}_{g,1}$, 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 ${\mathbb{M}}_{g,1}$. In particular, we construct a family of twisted differential forms on ${\mathbb{M}}_{g,1}$ representing the $(0,p+2)$-twisted Morita-Mumford class $m_{0,p+2}$ combinatorially parametrized by the Stasheff associahedron $K_{p+1}$.

38 pages

Cited by in corpus (7)

• Cohomological Aspects of Magnus Expansions
• Johnson's homomorphisms and the Arakelov-Green function
• The logarithms of Dehn twists
• The generalized Dehn twist along a figure eight
• Canonical 2-forms on the moduli space of Riemann surfaces
• Canonical lifts of the Johnson homomorphisms to the Torelli groupoid
• Surface topology and involutive bimodules