Timeline for p-local space vs p-completion
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 5, 2015 at 17:55 | comment | added | user62675 | @მამუკაჯიბლაძე No, I don't think there is. | |
| Sep 5, 2015 at 17:53 | comment | added | მამუკა ჯიბლაძე | Yes I understood that much. So there is no "correct" extension of that functor to all spaces? Or at least all CW-complexes... | |
| Sep 5, 2015 at 17:50 | comment | added | user62675 | @მამუკაჯიბლაძე There is a functor which takes a group to its $p$-completion. A map $X\to Y$ induces a map $L_{M(p)}X\to L_{M(p)}Y$, and you can take homotopy groups to get a map $\pi_\ast L_{M(p)}X\to \pi_\ast L_{M(p)}Y$. Now $\pi_\ast L_{M(p)}X\cong\pi_\ast X\otimes\mathbf{Z}_p$, so this is isomorphic to $\pi_\ast X^\wedge_p$ if $\pi_\ast X$ is finitely generated. Bousfield localization at $M(p)$ therefore gives you such a functor, but only on the subcategory of $\mathrm{Top}$ spanned by those spaces $X$ such that $\pi_\ast X$ is finitely generated. | |
| Sep 5, 2015 at 17:35 | comment | added | მამუკა ჯიბლაძე | Yes exactly. More precisely, is there a functor $F$ on spaces such that $\pi_*(F(X))=\pi_*(X)^\wedge_p$. | |
| Sep 5, 2015 at 17:24 | comment | added | user62675 | @მამუკაჯიბლაძე In general it does differ from $\pi_\ast X\otimes\mathbf{Z}_p$; if $\pi_\ast X$ is finitely generated, then then the two are isomorphic. I'm not sure I understand the first question; are you asking if there is a functor which takes a group to its $p$-completion? | |
| Sep 5, 2015 at 17:18 | comment | added | მამუკა ჯიბლაძე | Does any functor also yield $p$-completion of $\pi_*(X)$? Because in general it differs from $\pi_*(X)\otimes{\mathbf Z}_p$, right? | |
| Sep 5, 2015 at 17:03 | comment | added | user62675 | I realized that this has been said in the comments as well. | |
| Sep 5, 2015 at 16:38 | history | answered | user62675 | CC BY-SA 3.0 |