Timeline for Truncation and connected cover of spectra
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 26, 2017 at 6:55 | comment | added | Matthias Ludewig | Could somebody post this as an answer so I can accept? If anybody has a definite reference for these statements, I would greatly appreciate this! | |
| Sep 26, 2017 at 3:06 | comment | added | Nicholas Kuhn | Of course, then there are fun examples like $X = \mathbb RP^{\infty}$ and $Y = S$ (Segal conjecture for $\mathbb Z/2$) to mess with your head! (The mapping spectrum is still $-1$-connected.) | |
| Sep 26, 2017 at 1:00 | comment | added | Tyler Lawson | Just to tag along with Dylan's comments, if you want bounds on the nonzero groups of the function spectrum you most easily get them from bounds on the homotopy groups of Y and the cohomology groups of X. Connectivity can give you a bound on the cohomology of X but coconnectivity can't. | |
| Sep 25, 2017 at 21:11 | comment | added | Dylan Wilson | Yeah all your statements are okay with mapping space replacing [,]. (Obligatory comment about using derived mapping spaces if you're in some model category.) | |
| Sep 25, 2017 at 21:10 | comment | added | Dylan Wilson | The mapping spectrum will usually not be n-connective period. After all, the mapping spectrum into the sphere flips your spectrum upside down... or more readily: mapping spectra out of a sphere will give you desuspensions of the target, which are evidently less connective than they started. | |
| Sep 25, 2017 at 21:08 | comment | added | Matthias Ludewig | Sorry, I corrected my mistake. So the mapping spectrum between $X$ and $Y$ will generally not be connective unless both $X$ and $Y$ are connective? But he equalities do hold if I consider the mapping space? | |
| Sep 25, 2017 at 21:06 | history | edited | Matthias Ludewig | CC BY-SA 3.0 |
added 63 characters in body
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| Sep 25, 2017 at 21:04 | comment | added | Dylan Wilson | Yes, no (you've got it backwards), yes, and no (because mapping spectra know about maps from desuspensions of the source.) EDIT: Now you've got the second thing the right way round, so yes to that too. | |
| Sep 25, 2017 at 20:57 | history | asked | Matthias Ludewig | CC BY-SA 3.0 |