  Infinite presentations are given for all of the higher Torelli groups of
once-punctured surfaces. In the case of the classical Torelli group, a finite
presentation of the corresponding groupoid is also given, and finite
presentations of the classical Torelli groups acting trivially on homology
modulo N are derived for all N. Furthermore, the first Johnson homomorphism,
which is defined from the classical Torelli group to the third exterior power
of the homology of the surface, is shown to lift to an explicit canonical
1-cocycle of the Teichmueller space. The main tool for these results is the
known mapping class group invariant ideal cell decomposition of the
Teichmueller space.
  This new 1-cocycle is mapping class group equivariant, so various
contractions of its powers yield various combinatorial (co)cycles of the moduli
space of curves, which are also new. Our combinatorial construction can be
related to former works of Kawazumi and the first-named author with the
consequence that the algebra generated by the cohomology classes represented by
the new cocycles is precisely the tautological algebra of the moduli space.
  There is finally a discussion of prospects for similarly finding cocycle
lifts of the higher Johnson homomorphisms.
