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Infinite presentations are given for all of the higher Torelli groups ofonce-punctured surfaces. In the case of the classical Torelli group, a finitepresentation of the corresponding groupoid is also given, and finitepresentations of the classical Torelli groups acting trivially on homologymodulo N are derived for all N. Furthermore, the first Johnson homomorphism,which is defined from the classical Torelli group to the third exterior powerof the homology of the surface, is shown to lift to an explicit canonical1-cocycle of the Teichmueller space. The main tool for these results is theknown mapping class group invariant ideal cell decomposition of theTeichmueller space.  This new 1-cocycle is mapping class group equivariant, so variouscontractions of its powers yield various combinatorial (co)cycles of the modulispace of curves, which are also new. Our combinatorial construction can berelated to former works of Kawazumi and the first-named author with theconsequence that the algebra generated by the cohomology classes represented bythe new cocycles is precisely the tautological algebra of the moduli space.  There is finally a discussion of prospects for similarly finding cocyclelifts of the higher Johnson homomorphisms.

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves