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I will be very grateful for any advise or reference on the following.

1- How much is known about infinite families in ${_2\pi_*^s}$, the $2$-component of the stable homotopy ring?

2- How much is known about possible geometric construction of any of these families?!

I like to know about known constructions of such families by geometric methods. The geometric constructions that I am interested in are

(i) factoring an element of $f\in{_2\pi_*^s}$ through some finite dimensional complexes. The $\eta_i$ elements of Mahowald family are constructed in such a way. Also representing $f$ in terms of triple or higher Toda bracket will lead to such a factorisation, not necessarily unique.

ADDED I think Joel Cohen's result on representing elements of ${_2\pi_*^s}$ by (higher) Toda brackets, despite the question about the indeterminacy, means that any stable map $S^n\to S^0$ admits a factorisation through finite number of finite dimensional (stable) CW-complexes. For instance, the $\mu_i$ element in ${_2\pi_{8i+1}^s}$ coming from ${_2\pi_*}J$ with $J$ being the fibre of $\psi^3-1:BSO\to BSO$, is represented by a triple Toda bracket and by the construction of Adams, it factors through a $2$-cell complex.

(ii) constructing elements using homotopy operations arising as described by Bruner. For instance, Bruner's $\tau_i$ family is constructed using $\cup_1$ operation as described by Bruner.

I doubt if there is any structural result on the existence of such families; I presume whether or not if there exist finite number of such families is not known?! and if anything known would be a collection of latest results, something like what we find in Ravenel's Green book (do not know if a more updated reference exists!).

ADDED In particular, I like to know of any geometric construction of infinite families detected in the Adams or Adams-Novikov spectral sequences, such as those coming from Greek letter constructions? I must say that I dno't know much about the Greek Letter elements, so these might be very well documented somewhere in the literature. I am happy even to know about any conjectural construction or those which are folklore and believed to be true?!

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    $\begingroup$ Have you looked at "Mahowaldean families of elements in stable homotopy groups revisited" by Hunter and Kuhn? $\endgroup$ – Neil Strickland Jul 1 '16 at 16:39
  • $\begingroup$ @NeilStrickland Are you referring to this paper as a source of geometric constructions or are you referring to it as a place to look for a list of perhaps new infinite families? I have previously looked at this paper, although not for this purpose. I agree it is a good source for geometric constructions, but at least $\eta_i$ elements, at $p=2$, were known how to be constructed geometrically by Mahowald himself; maybe I have to reread it more carefully. I thought that at the prime $p=2$ they mostly reprove Mahowald's results using different methods, as well as odd primary Cohen's results. $\endgroup$ – user51223 Jul 1 '16 at 19:12
  • $\begingroup$ I have not properly digested the Hunter-Kuhn paper myself, and I don't make any specific claim about it except that it seems potentially relevant. $\endgroup$ – Neil Strickland Jul 3 '16 at 18:12
  • $\begingroup$ What I don't know much about, is the history of developments of the ideas in such problems. For instance, regarding homotopy operations and the Adams-Novikov SS, I don't know which was introduced first (I suspect ANSS), and on the other hand, how much is known about any possible geometric constructions of elements detected by ANSS and whether or not any of such constructions can be made using homotopy operations or Toda brackets, etc. I somehow felt that the history would provide more insight. $\endgroup$ – user51223 Jul 4 '16 at 15:24
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As you probably know, the existence of Greek letter elements relies on the existence of (generalized) Smith-Toda complexes -- the best introduction to those is probably still Section 1.3 of Ravenel's Green book, but Lee Nave's paper http://annals.math.princeton.edu/wp-content/uploads/annals-v171-n1-p10-p.pdf is certainly also work a look.

A generalized Smith-Toda complex is a finite complex whose $BP$-homology is $BP_*/(p^{i_0}, v_1^{i_1},\dots, v_n^{i_n})$ and it is often denoted by $M(i_0,\dots, i_n)$. They arise inductively by defining a self-map of $M(i_0, \dots, i_{n-1})$ that induces multiplication by $v_n^{i_n}$ on $BP$-homology and taking its cone. In general, the existence of such a $v_n$-self map $f$ is only known for some value of $i_n$ (by the periodicity theorem by Hopkins and Smith). The determination of a minimal $i_n$ is very difficult in general.

The basic procedure how these things give rise to periodic families in the stable homotopy groups of spheres is the following. Let's start with an element $x \in \pi_k M(i_0, \dots, i_{n-1})$. Then we can look at the composite $$S^k \to M(i_0,\dots, i_{n-1}) \xrightarrow{f^m} \Sigma^?M(i_0, \dots, i_{n-1}) \to S^?,$$ where the last map is "projection to the top cell". The value of this class in the Adams-Novikov spectral sequence can (usually) be explicitly computed although it can be a major issue to decide whether it is zero or not.

For example, $M(1)$ (i.e. the mod $p$ Moore spectrum) admits a $v_1^1$-self map for $p>2$, producing the $\alpha$-family; $M(1,1)$ admits a $v_2^1$-self map for $p>3$, producing the $\beta$-family; $M(1,1,1)$ admits a $v_3^1$-self map for $p>5$, producing the $\gamma$-family. Showing that the $\gamma$-family is indeed nonzero in the the Adams-Novikov spectral sequence was accomplished in the Miller-Ravenel-Wilson article Periodic Phenomena in the Adams-Novikov Spectral Sequence.

Note that none of this directly works for $p=2$. The mod-$2$ Moore spectrum $M(1)$ only admits a $v_1^4$-self map (see Adams's $J(X)$ IV). The resulting $M(1,4)$ admits only a $v_2^{32}$-self map. The latter result is proven in the Behrens-Hill-Hopkins-Mahowald paper https://www3.nd.edu/~mbehren1/papers/v2_32.pdf. The consequences for periodic families are discussed in Section 11 of the earlier preprint http://hopf.math.purdue.edu/Hopkins-Mahowald/eo2homotopy.pdf by Hopkins and Mahowald (although the account is rather condensed and I have never worked through it). For further background see theTMF-book and Homology of $tmf$ by Mathew.

There is a also later work on $v_2$-self maps at $p=2$ by Bhattacharya, Egger and Mahowald: http://arxiv.org/abs/1406.3297 -- I know nothing though about its consequences for periodic families.

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  • $\begingroup$ 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?!? $\endgroup$ – user51223 Jul 4 '16 at 14:59
  • $\begingroup$ 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 $\endgroup$ – Lennart Meier Jul 5 '16 at 8:29
  • $\begingroup$ 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). $\endgroup$ – Lennart Meier Jul 5 '16 at 8:34
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    $\begingroup$ 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. $\endgroup$ – Prasit Jul 8 '16 at 20:12
  • $\begingroup$ @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? $\endgroup$ – user51223 Jul 14 '16 at 13:17

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