Galatius and Randal-Williams proved the following generalized Mumford conjecture in their joint paper, ["*Stable Moduli spaces of High Dimensional Manifolds*"](https://arxiv.org/pdf/1201.3527.pdf). For each characteristic class of oriented $2n$-dimensional vector bundles $c\in H^{2n+k}(BSO(2n))$, one can define the associated generalized Mumford–Morita–Miller class of a smooth fiber bundle $\pi: E\to B$ with oriented
$2n$-dimensional fibers as $\kappa\_c(E)=\pi\_!(c(T\_vE))\in H^k(B),$
where $T\_v(E) = Ker(D\pi)$ is the fiberwise tangent bundle of $\pi$. When the fiber is taken to be $W\_g=\#^g S^n\times S^n$, the connected sum of $g$ copies of $S^n\times S^n,$ there is a corresponding universal class $\kappa\_c\in H^k(BDiff(W\_g, D^{2n}))$
which for $k>0$ is compatible with increasing $g$.


**Theorem** (Galatius-R-W) Let $2n>4$ and let $\mathcal{B}\subset H^\*(BSO(2n);\mathbb{Q})$ be the set of monomials
in the classes $e, p\_{n−1}, p\_{n−2},...,p\_{\left\lceil\frac{n+1}{4}\right\rceil}$ of total degree greater than $2n$. Then the
natural map $$\mathbb{Q}[\kappa\_c|c\in\mathcal{B}]=\lim\_{\longleftarrow g}H^\*(BDiff(W\_g,D^{2n});\mathbb{Q})$$ is an isomorphism.




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> I would like to determine the stable cohomology $$\lim\_{\longleftarrow g}H^\*(BDiff(W\_{g},D^{\infty});\mathbb{Q}).$$ Under the the isomorphism of G-R-W, this reduces to determining the MMM classes $\kappa\_c$ and the classes $e,p\_{\infty}$. How do I determine such classes? Moreover, is letting $n\to\infty$ in this construction permissible?
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Thanks!



Computing $H^*(BDiff(W_{\infty},D^{\infty});\mathbb{Q})$ via Mumford-Morita-Miller classes
As suggested by Dr. Hatcher, the dimension of the manifold $W\_g$, $2n$, is fixed and one considers embedded submanifolds of $\mathbb{R}^N$ diffeomorphic to $W\_g$. Then one lets $N$ go to infinity via the natural inclusion of $\mathbb{R}^N$ in $\mathbb{R}^{n+1}$, so submanifolds of $\mathbb{R}^N$ are regarded as submanifolds of $\mathbb{R}^{N+1}$. However the dimension $2n$ of the submanifolds is not changing. Thus, there is no natural way to let $n$ go to infinity in this construction.


