  Witten's class on the moduli space of 3-spin curves defines a
(non-semisimple) cohomological field theory. After a canonical modification, we
construct an associated semisimple CohFT with a non-trivial vanishing property
obtained from the homogeneity of Witten's class. Using the classification of
semisimple CohFTs by Givental-Teleman, we derive two main results. The first is
an explicit formula in the tautological ring of Mbar_{g,n} for Witten's class.
The second, using the vanishing property, is the construction of relations in
the tautological ring of Mbar_{g,n}. Pixton has previously conjectured a system
of tautological relations on Mbar_{g,n} (which extends the established
Faber-Zagier relations on M_g). Our 3-spin construction exactly yields Pixton's
conjectured relations. As the classification of CohFTs is a topological result
depending upon the Madsen-Weiss theorem (Mumford's conjecture), our
construction proves relations in cohomology. The study of Witten's class and
the associated tautological relations for r-spin curves via a parallel strategy
will be taken up in a following paper.

Algebraic Geometry