  We provide an intersection-theoretic formula for the Euler characteristic of
the moduli space of smooth curves. This formula reads purely in terms of Hodge
integrals and, as a corollary, the standard calculus of tautological classes
gives a new short proof of the Harer-Zagier formula. Our result is based on the
Gauss-Bonnet formula, and on the observation that a certain parametrisation of
the $\Omega$-class - the Chern class of the universal $r$-th root of the
twisted log canonical bundle - provides the Chern class of the log tangent
bundle to the moduli space of smooth curves. Being $\Omega$-classes by now
employed in many enumerative problems, mostly recently found and at times
surprisingly different from each other, we dedicate some work to produce an
extensive list of their general properties: extending existing ones, finding
new ones, and writing down some only known to the experts.
