Mumford's conjecture on the stable cohomology of mapping class groups

Let $\Gamma_g$ be the mapping class group of a closed oriented surface of genus $g$, and let $\kappa_1,\kappa_2,\ldots$ denote the Morita–Miller–Mumford classes. The tautological homomorphism of graded algebras is

$$\mathbb{Q}[\kappa_1,\kappa_2,\ldots]\longrightarrow H^*(B\Gamma_g,\mathbb{Q}).$$

Mumford's conjecture. This homomorphism should be an isomorphism in the stable range of Harer's stability theorem; in the formulation stated earlier, it is an isomorphism in degrees $*<g/2$.

Mumford formulated this conjecture to describe the stable rational cohomology of mapping class groups. The stated stable-range and degree-range assertions were proved by Madsen and Weiss.