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Feb 21, 2022 at 18:25 comment added Reid Barton See the last paragraph of mathoverflow.net/a/344958.
Feb 14, 2022 at 15:24 comment added Reid Barton I don't remember the details at the moment but I think it's fairly easy to deduce either of these statements (finite generation of $\pi_* S$ and $H\mathbb{Z}_* H\mathbb{Z}$) from the other--easier than proving either statement in the first place.
Feb 12, 2022 at 10:47 comment added Andrea Ferretti Thank you for your comment. Maybe @ReidBarton had in mind a different, somewhat "dual" proof? When I have time, I will check myself
Feb 11, 2022 at 17:35 comment added Wojowu Actually I don't think that's accurate. If you check out the proof in Scholze's notes, it uses finiteness not of homotopy group of spheres, but of homology groups of Eilenberg-MacLane spaces (which is a bit "dual": spheres have easy homology and hard homotopy, EM spaces have easy homotopy but hard homology). Those results apparently use very similar tools and methods, but are not equivalent. Commelin puts it as "The knowledge required is closely related to the finiteness of stable homotopygroups of spheres." (emphasis mine)
Feb 11, 2022 at 15:00 history answered Andrea Ferretti CC BY-SA 4.0