Explain the source of the super-exponential cohomology in Mg

Determine the nature of the super‑exponentially many rational cohomology classes in the moduli space of curves Mg implied by Harer–Zagier’s Euler characteristic formula; specifically, identify what classes in H*(Mg; Q) account for the lower bound dim H*(Mg; Q) ≥ (c · g)^{2g} for g ≥ 2.

Background

Harer and Zagier proved a formula for the Euler characteristic of Mg that yields a super‑exponential lower bound on the total dimension of H*(Mg; Q). Despite this, only comparatively small families of classes (e.g., tautological classes) had been known and these grow slower than exponentially with genus.

Chan, Galatius, and Payne related the top weight cohomology of Mg to commutative graph homology, and the present paper shows that the top weight piece itself grows super‑exponentially. Nevertheless, the overall Euler characteristic of Mg grows on the order of g^{2g}, which exceeds the growth detected by the top weight alone, leaving open the problem of identifying the classes responsible for the full super‑exponential growth.