Homological stability and Waldhausen A-theory

Question

$\DeclareMathOperator{\Diff}{Diff}$ 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?

Answer 1

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.