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 $K_{p+1}$.

arXiv comment38 pages

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• A tensorial description of the Turaev cobracket on genus 0 compact surfaces
• A combinatorial construction of symplectic expansions
• Canonical lifts of the Johnson homomorphisms to the Torelli groupoid
• Homotopy theory of $C_{\infty }$-algebras and characteristic classes of fiber bundles
• The center of the Goldman Lie algebra of a surface of infinite genus
• Surface topology and involutive bimodules
• Obstruction of $C_{\infty }$-algebra models and characteristic classes