The Madsen-Weiss Theorem, as described by Hatcher, states that there is an isomorphism $H\_\*( \mathscr{C}\_{\infty})\cong H\_\*(\Omega\_0^{\infty}AG^+\_{\infty,2})$ where $\Omega\_0^{\infty}AG^+\_{\infty,2}$ denotes the base point path-connected component of $\Omega^{\infty}AG^+\_{\infty,2}$ for the direct limit $\mathscr{C}\_{\infty}=\cup\_g\mathscr{C}\_g$ of $\mathscr{C}\_g$, the space of subsurfaces of $(-\infty,g]\times \mathbb{R}^{\infty}$ diffeomorphic to $S\_{g,1}$ (the compact surfaces of genus $g$ with one boundary component). 



> 
> However, is there an isomorphism which generalizes past $d=2$ using the scanning map $\mathscr{C}(M,\mathbb{R}^{\infty})\to\Omega^{\infty}AG^+\_{\infty,d}$? That is, is there an isomorphism involving $H\_\*(\mathscr{C}(M,\mathbb{R}^{\infty}))$ and $H^\*(\Omega\_0^{\infty}AG^+\_{\infty,d})$?
> 
> 
> 


Note $\mathscr{C}(M,\mathbb{R}^{\infty})$ denotes the space of all smooth oriented
submanifolds of $\mathbb{R}^{\infty}$ diffeomorphic to $M$ and $\Omega^{\infty}AG^+\_{\infty,d}$ is the limit of the $n$-fold loop space of the one-point compactification $AG^+\_{n,d}$ of $AG\_{n,d}$, the affine Grassmannian of
oriented flat $d$-planes in $\mathbb{R}^n$. 


I came across [this paper](https://arxiv.org/pdf/1203.6830.pdf), which I believe is the appropriate generalization if we replace the manifold $W\_{g,1}=\#^g(S^n\times S^n)-int(D^{2n})$ with $W\_{g,1}=\#^gS^n-int(D^n)$, although I am not certain.


See [Hatcher's paper](http://pi.math.cornell.edu/~hatcher/Papers/MW.pdf) for better reference.
Thanks in advance!



Generalizing the Madsen-Weiss Theorem via the scanning map $\mathscr{C}(M,\mathbb{R}^{\infty})\to\Omega^{\infty}AG^+_{\infty,d}$
The sequence of papers



> 
> S. Galatius, O. Randal-Williams, *Stable moduli spaces of high-dimensional manifolds*. Acta Math. 212 (2014), no. 2, 257–377. ([DOI: 10.1007/s11511-014-0112-7](https://doi.org/10.1007/s11511-014-0112-7), [projecteuclid](https://projecteuclid.org/euclid.acta/1485801812))
> 
> 
> S. Galatius, O. Randal-Williams, *Homological stability for moduli spaces of high dimensional manifolds. I*. J. Amer. Math. Soc. 31 (2018), no. 1, 215–264. ([arXiv: 1403.2334](https://arxiv.org/abs/1403.2334), [DOI: 10.1090/jams/884](https://doi.org/10.1090/jams/884))
> 
> 
> S. Galatius, O. Randal-Williams, *Homological stability for moduli spaces of high dimensional manifolds. II*. Ann. of Math. (2) 186 (2017), no. 1, 127–204. ([arXiv: 1601.00232](https://arxiv.org/abs/1601.00232), [DOI:annals.2017.186.1.4](https://doi.org/10.4007/annals.2017.186.1.4), [jstor](https://www.jstor.org/stable/26395664))
> 
> 
> 


addresses this question for even-dimensional manifolds. 


Combining the first and last papers shows that for *all* manifolds $M$ with non-empty boundary of even dimension $2n$ there is an analogous result to the Madsen--Weiss theorem, describing the homology of the moduli space of such manifolds after stabilising by $S^n \times S^n$, though the target of the scanning map must be modified to something adapted to $M$. 


The second paper shows a "homological stability theorem" for moduli spaces of simply-connected manifolds $M$ of dimension $2n \geq 6$, analogous to Harer's stability theorem for manifolds of dimension $2n=2$. This allows the conclusions of the other papers to also apply to e.g. closed manifolds in a range of degrees depending on the number of $S^n \times S^n$ connect-summands they contain. The paper



> 
> N. Friedrich, *Homological stability of automorphism groups of quadratic modules and manifolds*. Doc. Math. 22 (2017), 1729–1774. ([arXiv:1612.04584](https://arxiv.org/abs/1612.04584), [DOI: 10.25537/dm.2017v22.1729-1774](https://doi.org/10.25537/dm.2017v22.1729-1774))
> 
> 
> 


extends this to manifolds of dimension $2n \geq 6$ having virtually polycylic fundamental groups.


