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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, 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 for better reference. Thanks in advance!

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  • $\begingroup$ It might not be a generalization in the way you're looking for, but have you checked out the work of Galatius-Madsen-Tillmann-Weiss? $\endgroup$ Commented Aug 1, 2018 at 16:07
  • $\begingroup$ @ArunDebray No I haven't, thanks! Although, looking through it doesn't seem to be what I am looking for. $\endgroup$ Commented Aug 1, 2018 at 16:11

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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, projecteuclid)

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, DOI: 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, DOI:annals.2017.186.1.4, jstor)

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, DOI: 10.25537/dm.2017v22.1729-1774)

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

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  • $\begingroup$ Hello @OscarRandal-Williams! I am currently reading your paper, although I am not very far in. I have a question. I am interested in the case where we do have a connected sum of just $S^n$ and not $S^n\times S^n$, and are just concerned with $\mathscr{C}(M,\mathbb{R}^{\infty})$. Do you eventually show $H_*(\mathscr{C}(M,\mathbb{R}^{\infty}))\cong H^*(\Omega^{\infty}_0AG^+_{\infty,d})$? Could you perhaps tell me how this presentation and your argument would be modified for this case? $\endgroup$ Commented Aug 1, 2018 at 20:01
  • $\begingroup$ Connect summing with a sphere doesn't do anything: are you sure that is what you want? $\endgroup$ Commented Aug 1, 2018 at 20:04
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    $\begingroup$ @Multivariablecalculus: The Madsen-Weiss theorem does not compute the homology of BDiff of any particular surface, but only the stable value as the genus approaches infinity. The unstable homology is still basically completely unknown. Whatever you do, there will be some kind of stabilization involved. Galatius--Randal-Williams's results are the natural generalization of this. $\endgroup$ Commented Aug 1, 2018 at 21:55
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    $\begingroup$ Describing the homology of that space means at least classifying all 1-connected manifolds of a given dimension. You're not going to be able to do that. $\endgroup$ Commented Aug 1, 2018 at 21:56
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    $\begingroup$ $BDiff(\mathbb{C}^n) \simeq BO(2n)$, but determining the homotopy type of $BDiff(S^n)$ is a big open problem. $\endgroup$
    – skupers
    Commented Aug 2, 2018 at 1:19

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