Let $\text{Mod}\_g$ be the mapping class group of a closed oriented genus-$g$ surface $\Sigma\_g$ and let $H = H\_1(\Sigma\_g;\mathbb{Q})$. Fix some $r \geq 0$. It is known that the cohomology group $H^k(\text{Mod}\_g;H^{\otimes r})$ is independent of $g$ once $g$ is sufficiently large relative to $g$ and $r$. Does anyone know a concrete description of it?


For $r=0$, this is just the Madsen-Weiss theorem. For $r \geq 1$, this could be extracted from the paper


E. Looijenga,
Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map,
J. Algebraic Geom. 5 (1996), no. 1, 135-150.


However, this paper really answers a much more complicated question where you look at the cohomology in an irreducible algebraic representation of $\text{Sp}(2g,\mathbb{Q})$. You could assemble this to get information about $H^{\otimes r}$, but given how complicated Looijenga's answer is this would lead to something terrible. I'm hoping there is a reasonable closed-form answer for these specific representations.


I've worked through Looijenga's argument and extracted the following special case: if $k$ is even, then $H^k(\text{Mod}\_g;H) = 0$, while if $k$ is odd of the form $k = 2n-1$, then
$$H^k(\text{Mod}\_g;H) = \bigoplus\_{i=0}^{n-2} H^{2i}(\text{Mod}\_g;\mathbb{Q}).$$
Thanks to Dan Petersen in the comments for pointing out that I had originally screwed this up, as well as a related calculation for $r=2$.



Stable cohomology of mapping class group with coefficients in $H^{\otimes n}$
Appendix B of Randal-Williams' ["Cohomology of automorphism groups of free groups with twisted coefficients"](https://link.springer.com/article/10.1007/s00029-017-0311-0) gives a stable description of the graded $\mathbb{Q}[\Sigma\_q]$-module $H^\*(\Gamma\_g;H^{\otimes q})$.


