Timeline for K-theory of free $G$-sets and the classifying space, and generalization
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2020 at 20:18 | history | edited | LSpice | CC BY-SA 4.0 |
Link to paper
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| Jan 19, 2020 at 13:44 | comment | added | John Klein | @user43326 even with the notation you say you like to use (and which I don't), it would have been more correct to write $BG_+$ rather than $BG$. | |
| Jan 19, 2020 at 9:18 | comment | added | Neil Strickland | @JohnKlein different people use different notation. I would quite often leave the $\Sigma^\infty$ implicit in this kind of context. | |
| Jan 19, 2020 at 8:08 | comment | added | user43326 | Oh, thank you, I had missed that. In most of my writing, I identify a based space with its suspension spectrum... Corrected now. | |
| Jan 19, 2020 at 0:20 | comment | added | John Klein | The expression "$BG$" refers to a space, not a spectrum. You are using incorrect notation. If you want to associate a spectrum to $BG$, you should be writing $\Sigma^\infty (BG_+)$--the suspension spectrum of $BG$. Again, I am saying there is an incorrect statement in the first paragraph of your post. | |
| Jan 17, 2020 at 15:15 | comment | added | user43326 | that is exactly what is I am saying, for some authors it is the spectrum and not the infinite loop space... | |
| Jan 17, 2020 at 12:16 | comment | added | John Klein | @user43326: every connective spectrum determines an infinite loop space and vice-versa. | |
| Jan 17, 2020 at 12:14 | comment | added | John Klein | @user43326 the K-theory of finite free $G$-sets is not $BG$, it is $Q(BG_+)$. Here is a simple reason why it can't be $BG$: the latter is not in general an infiinite loop space (but the $K$-theory is an infinite loop space). By the way, $G$ does not need to be finite in the Barratt-Priddy-Quillen-Segal theorem. | |
| Jan 17, 2020 at 6:48 | comment | added | user43326 | Actually I really meant $BG_+$ because some author use the word $k$-theory to mean the associated spectrum (infinite delooping) and not the infinite loop space. | |
| Jan 17, 2020 at 2:10 | history | answered | John Klein | CC BY-SA 4.0 |