Timeline for Group Completions and Infinite-Loop Spaces
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
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| Sep 1, 2010 at 13:16 | vote | accept | Lennart Meier | ||
| Sep 1, 2010 at 13:16 | history | bounty ended | Lennart Meier | ||
| Aug 29, 2010 at 19:56 | comment | added | Lennart Meier | Actually I have a stable symmetric monoidal $\infty$-category (in the sense of Joyal-Lurie). But I don't know if this has any relevant information for this question beyond the informations already stated. The example of the $B\Sigma_n$ perhaps just felt wrong. I must admit it is a "counterexample" to the conditions I have stated. | |
| Aug 29, 2010 at 18:01 | comment | added | Tyler Lawson |
I think there may be some unstated assumption remaining, then? From Torsten's answer, the loop space of $K(\Sigma_n,1)$ is an infinite loop space, just not in a way compatible with the loop-space structure.
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| Aug 29, 2010 at 15:34 | comment | added | Lennart Meier | I may have been unclear again: I don't assume the path-components to be infinite-loop spaces but their loop spaces. Is this enough to deduce them to be simple? | |
| Aug 29, 2010 at 14:08 | comment | added | Tyler Lawson |
Ah. This may actually be simpler than I initially thought. If $X$ is an $E_\infty$-space, in general one group-completes by taking a homotopy colimit along multiplication maps (which localizes the homology), and then applies a plus-construction. However, if all the path-components are infinite loop spaces, they are already simple and so one can calculate the higher homotopy groups as just those obtained from the homotopy colimit - is that along the lines of what you wanted?
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| Aug 29, 2010 at 12:58 | comment | added | Lennart Meier | I'm taking this back. The situation where I actually am is the following: all occuring loop spaces are the infinite loop spaces associated to $A_\infty$-ring spectra. On the identity component, the $E_\infty$-structure coming from X should be the same as that from the infinite-loop structure (although the identity component is boring in my case). In the other components, the loop multiplication should be compatible with the $A_\infty$-multiplicative structure. | |
| Aug 29, 2010 at 12:35 | comment | added | Tyler Lawson |
How do you obtain an $E_\infty$-structure on the loop space based at a nonidentity point?
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| Aug 29, 2010 at 10:06 | comment | added | Lennart Meier | I should add that I want the $E_\infty$-structure on $\Omega(X,x)$ coming from the infinite-loop space structure to be compatible with that coming from X. | |
| Aug 29, 2010 at 7:50 | comment | added | Lennart Meier | @Tyler: Eg the loop space based at the identity. More precisely: for every $x \in X$, the loop space $\Omega(X,x)$ is an infinite loop space. | |
| Aug 29, 2010 at 6:54 | answer | added | Torsten Ekedahl | timeline score: 11 | |
| Aug 28, 2010 at 22:16 | comment | added | Dan Ramras | My feeling is that in order for the answer to (1) to be "Yes" you should allow local coefficient systems in homology. Does anyone have a more precise thought about this point? | |
| Aug 28, 2010 at 17:21 | comment | added | Tyler Lawson | Question (1) leads me to a many more questions. Like: suppose we have a cofibrant simplicial object in unital monoids and we levelwise adjoin inverses; is that a "group completion" in this sense of localizing homology? This is much harder when you can't play games with the classifying space. Re question (3): I'm not sure what you mean by "all loop spaces of X are infinite loop spaces" - do you mean e.g. that the loop space based at the identity is an infinite loop space? | |
| Aug 28, 2010 at 11:51 | history | bounty started | Lennart Meier | ||
| Aug 26, 2010 at 13:02 | history | edited | André Henriques | CC BY-SA 2.5 |
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| Aug 26, 2010 at 12:33 | history | edited | Lennart Meier | CC BY-SA 2.5 |
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| Aug 25, 2010 at 13:48 | history | asked | Lennart Meier | CC BY-SA 2.5 |