topological stable rank one and AF-algebra construction on Cantor set

at.algebraic topology – Homological stability and Waldhausen A-theory Answer

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at.algebraic topology – Homological stability and Waldhausen A-theory

From the labor of Galatius – Randall-Williams and Berglund – Madsen we have now homological stability (with respect to g) of $BDiff_partial (W_{g,1})$ and rational homological stability of $Bwidetilde{Diff}_{partial}(W_{g,1})$, the place $W_{g,1}$ is $(#_g S^d instances S^d ) setminus D^{2nd}$ and $widetilde{Diff}$ denotes hindrance diffeomorphisms (primarily an analogue of diffeomorphisms the place path elements are naturally pseudoisotopy courses).

By analyzing the Serre spectral sequence for the fibration $widetilde{Diff}_{partial}(W_{g,1}) / Diff_partial (W_{g,1}) rightarrow BDiff_partial (W_{g,1}) rightarrow Bwidetilde{Diff}_{partial}(W_{g,1}) $ we deduce from the aforementioned homological stability outcomes, that $H_*(widetilde{Diff}_{partial}(W_{g,1}) / Diff_partial (W_{g,1});mathbb{Q})^{pi_0(widetilde{Diff}_{partial}(W_{g,1}))} $ has homological stability with respect to g. By pseudo-isotopy implies isotopy and a few of Wall’s labor on extremely linked manifolds, one might supplant $pi_0(widetilde{Diff}_{partial}(W_{g,1}))$ by a inescapable arithmetic group $Gamma$ if she wished.

It is understood that $widetilde{Diff}_{partial}(W_{g,1}) / Diff_partial (W_{g,1})$ is expounded to Waldhausen A-theory, behold Weiss’ and Williams’ Automorphisms of manifolds and algebraic Okay-theory: I. Is there a recognized proof utilizing A-theory that the invariants of those homology teams have homological stability?

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