An intersection-theoretic proof of the Harer-Zagier formula

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 Ω-class – the Chern class of the universal rth root of the twisted log canonical bundle – provides the Chern class of the log tangent bundle to the moduli space of smooth curves. These Ω-classes have been recently employed in a great variety of enumerative problems. We produce a list of their properties, proving new ones, collecting the properties already in the literature or only known to the experts, and extending some of them