Infinite families in stable homotopy groups

Question

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?!

Answer 1

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.