Timeline for $(B\mathbb Z/p^{\infty})^{\wedge}_p\rightarrow (BS^1)^{\wedge}_p$ induced by inclusion is an equivalence

7 events

when toggle format	what		by	license	comment
Sep 29, 2020 at 22:22	vote	accept	Victor TC
Sep 4, 2020 at 18:02	comment	added	Victor TC		Fine!, I was wrong, sorry. Thank you again.
Sep 4, 2020 at 5:47	comment	added	Achim Krause		I'm not sure I correctly remember what $p$-good means, but note that the mod $p$ homology of $B\mathbb{Q}$ is of course zero.
Sep 3, 2020 at 22:38	comment	added	Victor TC		but $B\mathbb{R}$ is contractible, and $B\mathbb{Q}$ is $p$-good. So far, their $p$-completions cannot be homotopy equivalent.
Sep 3, 2020 at 7:30	comment	added	Achim Krause		Yes, I think if you want to turn this idea into an actual proof you look at the induced map of fiber sequences $B\mathbb{Q}\to B\mathbb{Q/Z} \to K(Z,2)$ and $B\mathbb{R}\to B\mathbb{S^1}\to K(Z,2)$, and observe that the map on the fibers becomes an equivalence after $p$-completion.
Sep 3, 2020 at 1:28	comment	added	Victor TC		Thank you!, do you mean $B\mathbb{Q}^{\wedge}_p\rightarrow B\mathbb{R}^{\wedge}_p$ is a homotopy equivalence?
Sep 2, 2020 at 22:50	history	answered	Achim Krause	CC BY-SA 4.0