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?!