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Jul 15, 2016 at 10:52 comment added user51223 @Prasit Thanks for the reference to a proof of the point in that remark. I got the numbers slightly wrong! Above, you considered $M(1,4)$ where that Remark 1.4 in that paper is about $M(2,4)$.
Jul 15, 2016 at 6:39 comment added Prasit @Lennart Meier: For $X= M(1,4)$ or $X = A_1$, the infinite family looks like this $$ S^{192n} \to \Sigma^{192n} X \overset{v_2^{32n}}{\to} sk_{l_n} X \to S^{l_n}$$ where sk_{l_n} is the $l_n$-skeleton and $l_n$ depends on $n$ as well as $X$. There is apriori no reason to believe that $l_n$ match up for $A_1$ and $M(1,4)$. Determining $l_n$ is a hard problem. But if I have to make a guess then I would guess $l_n = 1$ for all $n$ and $X=M(1,4)$ and $X=A_1$ and they produce the same family. I could easily be wrong about this.
Jul 15, 2016 at 2:40 comment added Prasit @user51223: Remark 1.4 is a Conjecture that reports that just like M(1,4), there are type 2 complexes named $A_1$ and $M(2,4)$ which admit v2^32 self-map which is minimal ie $v_2^{16}$ or $v_2^8$ does not exist as selfmaps. The proof for $A_1$ is given by myself, P.Egger and M.Mahowald. IS that what you were wondering?
Jul 14, 2016 at 13:27 comment added user51223 @LennartMeier If my guess on identifying a given infinite family with a Greek letter construction, then is there any reference to this or does this follow from the construction of Morava $K$-theories and the philosophy behind it? So, for example does Mahowald's family admit such a construction?
Jul 14, 2016 at 13:17 comment added user51223 @Prasit I just happen to go back and look at a paper of Behrens-Hill-Hopkins-Mahowald, and I wonder if this is in the same of Remark 1.4 of that paper?
Jul 8, 2016 at 20:12 comment added Prasit The 32-periodic v2-selfmap on A1 and M(1,4) are may or may not produce the same periodic family. Recently, in a joint work with Egger, we produce a finite spectrum which has 1-periodic v2-selfmap at prime 2 which for sure is going to produce 'less sparse', hence different, v2-periodic family.
Jul 5, 2016 at 8:34 comment added Lennart Meier Suppose you have defined $M(i_0,\dots, i_n)$ as the cone of a $v_n^{i_n}$-self map of $M(i_0,\dots, i_{n-1}$. Then we obtain a map $M(i_0,\dots, i_n) \to \Sigma^{i_n|v_n|+1}M(i_0,\dots, i_{n-1})$. This defines inductively a map $M(i_0,\dots, i_n) \to S^{i_1|v_1|+\cdots i_n|v_n| + (n+1)}$. This is the "top cell"; the bottom cell is in dimension $0$. It is also true that $M(i_0,\dots, i_n)$ is a space after a finite number of suspension as this is true for every finite spectrum (i.e. every spectrum that is built by a finite number of cones from the sphere spectrum).
Jul 5, 2016 at 8:29 comment added Lennart Meier I do not think that you have uniqueness generally, but there is some "stable" uniqueness of the $v_n$-self maps. See for example Thm 1.5.4 of Ravenel's "orange book" math.rochester.edu/people/faculty/doug/mybooks/nilpb.pdf
Jul 4, 2016 at 14:59 comment added user51223 This is very informative. I think a spectrum $M(i_0,\ldots,i_n)$ is not necessarily unique, right! I also wonder if $M(i_0,\ldots,i_n)$ can be realised as a space after finite number of suspensions?!? I wonder how much is known/can be said about the CW-structure of this spectrum, and in particular about the dimension of its top and bottom cells?!?
Jul 4, 2016 at 10:26 history answered Lennart Meier CC BY-SA 3.0