Timeline for Connectivity of suspension-loop adjunction
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Sep 5, 2018 at 17:54 | history | edited | j.c. | CC BY-SA 4.0 |
fix typo
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| Sep 5, 2018 at 15:03 | vote | accept | Niall Taggart | ||
| Sep 5, 2018 at 14:26 | answer | added | Fernando Muro | timeline score: 10 | |
| Sep 5, 2018 at 13:51 | comment | added | Fernando Muro | @JohnKlein sure, I didn't claim my bound was optimal, it's obviously not optimal for $k=0$ since $\Omega^\infty X$ has abelian fundamental group. Actually, my argument above shows that the map is $(2k+1)$-connected. Let me provide an alternative answer below with the same bound as you get. | |
| Sep 5, 2018 at 13:14 | comment | added | John Klein | @FernandoMuro See my answer below. | |
| Sep 5, 2018 at 11:55 | comment | added | John Klein | @FernandoMuro: consider the case when $X$ is a suspension spectrum and use the Snaith splitting. You will see that I am right in that case. Alternatively, compute the Taylor tower of the functor $X\mapsto \Sigma^\infty \Omega^\infty X$. It has $j$-th layer $D_jX$ for $k \ge 1$. Note that $D_jX$ has connectivity $jr+j-1$ if $X$ is $r$-connected. If $j \ge 2$, then it is at least $2r+1$-connected. This will imply that the map from the top to the bottom of the tower is $(2r+2)$-connected. This is a high powered argument. I am trying to construct an elementary one. | |
| Sep 5, 2018 at 11:51 | comment | added | John Klein | If the spectrum $X$ is $r$-connected, then the map $\Sigma^\infty\Omega^\infty X \to X$ is $(2r+2)$-connected. | |
| Sep 5, 2018 at 11:41 | comment | added | Fernando Muro | @JohnKlein could be, but I don't think so. Freudenthat's suspension theorem shows that the homotopy groups of $\Sigma^\infty\Omega^\infty X$ coincide up to dimension $2k$ with those of $\Omega^\infty X$, which in turn are those of $X$. Hence my comment. | |
| Sep 5, 2018 at 11:31 | comment | added | John Klein | @FernandoMuro I think you are slightly off. | |
| Sep 5, 2018 at 11:30 | answer | added | John Klein | timeline score: 11 | |
| Sep 5, 2018 at 11:07 | comment | added | Fernando Muro | For $k\geq 0$, it is $2k$-connected by Freudenthal's suspension theorem. | |
| Sep 5, 2018 at 11:06 | history | edited | Niall Taggart | CC BY-SA 4.0 |
added 20 characters in body
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| Sep 5, 2018 at 10:36 | history | asked | Niall Taggart | CC BY-SA 4.0 |