Timeline for Group Completions and Infinite-Loop Spaces
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 1, 2010 at 13:16 | vote | accept | Lennart Meier | ||
| Sep 1, 2010 at 13:16 | history | bounty ended | Lennart Meier | ||
| Aug 30, 2010 at 4:19 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Comments on homotopy limits.; added 4 characters in body
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| Aug 29, 2010 at 13:17 | comment | added | Torsten Ekedahl | @Tyler: Thanks you are right. (I could possibly weasel my way out of that by claiming that what is needed is the associativity of the Pontryagin product which seems to be implied by the OP's use of "Pontryagin ring".) | |
| Aug 29, 2010 at 13:15 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Added a comment on associativity.
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| Aug 29, 2010 at 12:26 | comment | added | Tyler Lawson | @Torsten: Nice answer. One minor point is that you need to make the assumption that X is homotopy-associative (or possibly something weaker) in order to describe the localization as having homology obtained by tensoring up the base ring. | |
| Aug 29, 2010 at 11:03 | comment | added | Torsten Ekedahl | The map is obtained by obstruction theory (i.e., by induction over a Postnikov tower), see for instance Adams for a more detailed discsussion. I'll have to think about the other two parts. | |
| Aug 29, 2010 at 10:26 | comment | added | Lennart Meier | Ad 3): I know that the homotopy groups of group completions in general can go wild (else algebraic K-theory would behave quite differently!). But note that the $\Sigma_n$ are no infinite-loop spaces and therefore do not fit the conditions I posed (which are hopefully clearer after my comments to the question). | |
| Aug 29, 2010 at 10:25 | comment | added | Lennart Meier | Thanks for your answer! Ad 1): I see that I can apply this variant of Whitehead if I have a map. Where do I get the map from? Ad 2): I meant homotopy limits. If I haven't miscalculated, group completion should commute with products, at least under suitable flatness conditions. I'm actually interested mostly in a homotopy limit over a cosimplicial diagram. Perhaps one can use here the homology spectral sequence for a cosimplicial space. I need to check. | |
| Aug 29, 2010 at 6:54 | history | answered | Torsten Ekedahl | CC BY-SA 2.5 |