$\DeclareMathOperator{\Diff}{Diff}$ From the work of Galatius - Randall-Williams and Berglund - Madsen we have homological stability (with respect to g) of $B\Diff_\partial (W_{g,1})$ and rational homological stability of $B\widetilde\Diff_\partial(W_{g,1})$, where $W_{g,1}$ is $(\#_g S^d \times S^d ) \setminus D^{2d}$ and $\widetilde\Diff$ denotes block diffeomorphisms (essentially an analogue of diffeomorphisms where path components are naturally pseudoisotopy classes).

By analyzing the Serre spectral sequence for the fibration $\widetilde\Diff_\partial(W_{g,1}) / \Diff_\partial (W_{g,1}) \rightarrow B\Diff_\partial (W_{g,1}) \rightarrow B\widetilde\Diff_\partial(W_{g,1}) $ we deduce from the aforementioned homological stability results, that $H_*(\widetilde\Diff_\partial(W_{g,1}) / \Diff_\partial (W_{g,1});\mathbb Q)_{\pi_0(\widetilde\Diff_\partial(W_{g,1}))} $ has stability with respect to $g$. By pseudo-isotopy implies isotopy and some of Wall's work on highly connected manifolds, one could replace $\pi_0(\widetilde\Diff_\partial(W_{g,1}))$ by a certain arithmetic group $\Gamma$ if she wished.

It is known that $\widetilde\Diff_\partial(W_{g,1}) / \Diff_\partial (W_{g,1})$ is related to Waldhausen A-theory, see Weiss' and Williams' Automorphisms of manifolds and algebraic K-theory: I. Is there a known proof using A-theory that the coinvariants of these homology groups have stability?

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    $\begingroup$ I think I haven't seen the suggested statement explicilty stated as in your post before, but it is definitely in the spirit of very recent research. For instance, are you aware of this preprint by M. Krannich? arxiv.org/pdf/2002.04647.pdf $\endgroup$ Jan 10, 2021 at 11:32
  • $\begingroup$ @JensReinhold Thanks Jens, it looks like he's interested in essentially the same thing I am, but for odd dimensions; this will be good to compare with $\endgroup$ Jan 10, 2021 at 20:04

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I don't think that you can deduce homological stability of the coinvariants from the Serre spectral sequence as you suggest. But this precise situation was studied in my paper "An upper bound for the pseudoisotopy stable range", which may be useful.

To answer your last question, A-theory only describe these groups in a range of degrees depending on the dimension of the manifold, so could not be used to prove homological stability of these coinvariants in a range of degrees tending to infinity. In the range of degrees in which this approximation applies the coinvariants are in any case zero, so do indeed stabilise.

  • $\begingroup$ I could definitely be incorrect in my deduction. I was basing it off arguments I had seen which have applied “spectral sequence comparison” theorems to deduce homological stability. My understanding was that two of the three imply the third: isomorphisms of the zeroth row of the $E_2$ page in a large range, isomorphisms of the zeroth column of the $E_2$ page in a large range, and isomorphisms of the $E_\infty$ page in a large range. $\endgroup$ Jan 10, 2021 at 15:11
  • $\begingroup$ Do you know a correct statement of the type of result I am referring to? $\endgroup$ Jan 10, 2021 at 15:12
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    $\begingroup$ For Zeeman's comparison theorem you need the E_2-page to have a product structure, which it typically will not if the fundamental group of the base acts nontrivially on the homology of the fibre. (The formulation on Wikipedia is wrong.) $\endgroup$ Jan 10, 2021 at 16:18

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