Timeline for Homology of a limit of spectra + Cofiber
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 8, 2017 at 16:39 | comment | added | Tyler Lawson | @Alfred Again, I'm still not sure that you can ensure that the $d_n$ aren't forced to go to $\infty$ in that case. For example, when $n=2$ you are looking for Moore spaces $M(p^k,d)$ with $v_1^l$-self-maps -- with $k$ independent of $d$ -- and even if you can accomplish that, I'm not sure if you can get $v_2^m$-self-maps uniformly on the results, because the powers $m$ you must choose grow as $k$ and $l$ do. However, I don't know much unstable chromatic theory. | |
| Dec 7, 2017 at 20:08 | comment | added | Alfred | Thank you again! Last question, promise: Do you think that is there a way to define these spectra such that this cofiber has at least one non torsion element? What i can think is a way to make $BP_k(X_n) \neq 0$ for almost every $n$ by having a double-indexed sum $\bigvee X_{n,i}$ such that $BP_*(X_{n,i})= \Sigma^{d_n} BP_* / (V_0^{k_{0,n}}, \dots, v_n^{k_{n,n}})$, where every $k_j > i$ | |
| Dec 7, 2017 at 19:40 | comment | added | Tyler Lawson | @Alfred The existence of finite spectra $X_i$ like you described is guaranteed by results of Devinatz-Hopkins-Smith, and you can read about this in Ravenel's Nipotence and periodicity in stable homotopy theory (the "orange book"). But to get non-torsion you need the $d_n$ to not go to $\infty$ while maintaining the property of being suspension spectra and I cannot guarantee that: in fact, I suspect that it might be false. | |
| Dec 7, 2017 at 19:08 | comment | added | Alfred | I've written my question in the wrong way. What i meant was: Is it true that for every $i \geq 0$ there exist a finite (and then, with the proper $d_n,$ suspension) spectrum $X_i$ with the properties you wrote in the "ultrafilter" case, so with the $k_{n,i}$ going to infinity? | |
| Dec 7, 2017 at 17:20 | comment | added | Alfred | Thank you very much for the clear answer! Looks that this ultrafilter argument may be what i'm looking for, could you please suggest me a text-paper-book where to find something more about it? For example, the spectrum you mentioned with the $k_{n,i}$ going to infinity, is still finite and suspension? Thanks!! | |
| Dec 7, 2017 at 17:07 | vote | accept | Alfred | ||
| Dec 7, 2017 at 15:59 | history | answered | Tyler Lawson | CC BY-SA 3.0 |