Timeline for $K_3(\mathbb{Z})$ and $\pi ^S_3$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 31, 2019 at 14:04 | comment | added | John Rognes | The map to the cokernel of $\pi_3(S) \to K_3(\mathbb{Z})$ can be interpreted as the Hurewicz homomorphism to $H_3(K(\mathbb{Z})) \cong \mathbb{Z}/2$, or as the Bökstedt trace map to $\pi_3 THH(\mathbb{Z}) \cong \mathbb{Z}/2$. A key point in the 1976 paper by Lee and Szczarba is why the extension is nontrivial. I use $\lambda$ to denote a generator of $K_3(\mathbb{Z})$ to refer to Lee (and Szczarba). | |
| Aug 31, 2019 at 13:28 | vote | accept | abx | ||
| Aug 31, 2019 at 11:54 | answer | added | Drew Heard | timeline score: 9 | |
| Aug 31, 2019 at 9:56 | comment | added | David Roberts♦ | @abx ah, but which is the image of index 2? Is there some cokernel with another interpretation? | |
| Aug 31, 2019 at 9:35 | comment | added | abx | @Denis Nardin: Thanks a lot! Indeed Mitchell's paper completely answers my question. Sorry if it was too elementary, this is not my field. | |
| Aug 31, 2019 at 9:23 | comment | added | Denis Nardin | One quick way of constructing it is as the map induced on K-theory by the exact functor of Waldhausen categories $\mathrm{Fin}_*\to\mathrm{Proj}_{\mathbb{Z}}$ sending $1_+$ to $\mathbb{Z}$ | |
| Aug 31, 2019 at 9:15 | comment | added | Denis Nardin | @abx It's just the unit of the ring structure. I think pretty much any book on algebraic K-theory constructs it, you can see my answer here for a few references (I like a lot Mitchell's survey in that list) | |
| Aug 31, 2019 at 9:00 | comment | added | abx | @ Achim Krause: Could you explain where this map comes from (or give a reference)? | |
| Aug 31, 2019 at 8:43 | history | edited | YCor |
edited tags
|
|
| Aug 31, 2019 at 8:40 | comment | added | Achim Krause | There is a map of spectra $\mathbb{S}\to K(\mathbb{Z})$. I'm not completely sure, but I'd expect it to be injective on $\pi_3$. | |
| Aug 31, 2019 at 8:07 | history | asked | abx | CC BY-SA 4.0 |