Alex James Bene

We prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized Magnus expansion in the sense of Kawazumi. This Magnus expansion is used to give canonical lifts of the higher Johnson homomorphisms $\tau_m$, for $m\geq 1$, to the Torelli groupoid, and we provide a recursive combinatorial formula for tensor representatives of these lifts. In particular, we give an explicit 1-cocycle in the dual fatgraph complex which lifts $\tau_2$ and thus answer affirmatively a question of Morita-Penner. To illustrate our techniques for calculating higher Johnson homomorphisms in general, we give explicit examples calculating $\tau_m$, for $m\leq 3$.

A closed formula is obtained for the integral $\int_{\mathcal{\bar{H}}_g^1}\kappa_{1}\psi^{2g-2}$ of tautological classes over the locus of hyperelliptic Weierstra\ss{} points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained. The calculation utilizes the homeomorphism between the moduli space of curves $\mathcal{M}_{g,1}$ and the combinatorial moduli space $\mathcal{M}^{comb}_{g,1}$, a PL-orbifold whose cells are enumerated by fatgraphs. This cell decomposition can be used to naturally construct combinatorial PL-cycles $W_a\subset\mathcal{M}^{comb}_{g,1}$ whose homology classes are essentially the Poincar\'e duals of the Mumford-Morita-Miller classes $\kappa_a$. In this paper we construct another PL-cycle $\mathcal{H}^{comb}_g \subset \mathcal{M}^{comb}_{g,1}$ representing the locus of hyperelliptic Weierstra\ss{} points and explicitly describe the chain level intersection of this cycle with $W_1$. Using this description of $\mathcal{H}^{comb}_g\cap W_1$, the duality between Witten cycles $W_a$ and the $\kappa_a$ classes, and Kontsevich's scheme of integrating $\psi$ classes, the integral $\int_{\mathcal{\bar{H}}_g^1}\kappa_{1}\psi^{2g-2}$ is reduced to a weighted sum over graphs and is evaluated by the enumeration of trees.