Timeline for What can be said about the map $K(n)_\ast(X) \to K(n)_\ast(\tau_{\leq n}X)$ when $X$ is a finite complex?
Current License: CC BY-SA 4.0
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| Apr 18, 2019 at 0:42 | answer | added | Nicholas Kuhn | timeline score: 6 | |
| Apr 17, 2019 at 23:50 | comment | added | Tim Campion | er -- $\tau_{\geq n+1}$ should be $\tau_{\geq n+2}$ and $m \geq n$ should be $m \geq n+1$ in my comment | |
| Apr 17, 2019 at 23:49 | history | rollback | Tim Campion |
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| Apr 17, 2019 at 23:26 | comment | added | Tim Campion | @EricPeterson It seems to me that the analogous statement for unstable connective covers is that $\tau_{\geq m} X \to \tau_{\geq n+1} X$ is a $K(n)_\ast$-equivalence, but this doesn't extend to lower connective covers. Instead, the sequence $\tau_{\geq m+1} X \to X \to \tau_{\leq m} X$ (which is the same for all $m \geq n$) is a kind of fundamental decomposition of $X$, both parts of which are potentially interesting -- though as HRW point out in the paper you linked to, there's a strong tendency for the truncated part to split uninterestingly. | |
| Apr 17, 2019 at 23:14 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 17, 2019 at 22:16 | comment | added | Nicholas Kuhn | @NeilStrickland Neil and others: I wrote a long paper on the Morava K-theory of infinite loopspaces [Adv. Math. 201 (2006), 318-378]. It is quite definitive, and, yes, does use the $\Phi_n$ functors. Folks are encouraged to look at this to get a sense of weird stuff that can happen. | |
| Apr 17, 2019 at 21:44 | comment | added | Neil Strickland | I don't know a good answer, but I suspect that this could be a fruitful area to investigate. One possible approach: the above comments tell us about $K(n)_*(\Omega^\infty X)$ for $X$ in the thick subcategory generated by $H=BP\langle 0\rangle$, and some of the proofs can be given in terms of the Bousfield-Kuhn functor $\Phi_n$ which satisfies $\Phi(\Omega^\infty X)=L_{K(n)}X$. So you can proceed to investigate $K(n)_*(\Omega^\infty X)$ when $X$ is in the thick subcategory generated by $BP\langle m\rangle$ with $m>0$, using the theory of Wilson spaces. | |
| Apr 17, 2019 at 20:31 | comment | added | Eric Peterson | You can make pretty strong statements about what $K(n)_* \tau_{\le k} X$ looks like, though, which you might find separately interesting: math.jhu.edu/~wsw/papers2/math/… . | |
| Apr 17, 2019 at 20:31 | comment | added | Eric Peterson | I wouldn't think that there is any meaningful relationship. For $E$ stable, there is an equivalence $L_{K(n)} \tau_{\ge k} E \simeq L_{K(n)} E$, so that all of the information that $L_{K(n)}$ (or $K(n)_*$) sees is attached to the "germ at infinity" of $E$ (a memorable slogan credited to Dwyer). Since the parts of $\Sigma^\infty_+ X$ and $\Sigma^\infty_+ \tau_{\le k} X$ near infinity can be arbitrarily far apart without very strong hypotheses on $X$, their $K(n)$-homologies are not likely to bear any particular relation. | |
| Apr 17, 2019 at 18:22 | comment | added | Denis Nardin | You do have the Serre Spectral sequence, but I'm not sure how helpful it is | |
| Apr 17, 2019 at 18:16 | history | asked | Tim Campion | CC BY-SA 4.0 |