3 -periodic classes
Next, we look at the orange and yellow classes. The yellow classes is a free -module on a single generator , where . There is no actual element that you can multiply with; rather, it is given by the Massey product .
However, some multiples of are realized by elements in , namely
So exists for and is permanent for sufficiently large . These yellow classes are in fact all permanent.
It is useful to note that is in fact itself divisible, with . Note that is the sum of the two basis elements in bidegree . The two basis elements can be uniquely identified as follows — the brown class is the unique non-zero class that is torsion, while the black class is the unique non-zero class that is divisible.
These classes stay along for quite a while. The differentials that eventually kill them come from the differential
in the Adams spectral sequence for .
The orange classes form a free -module on a single generator . Again exists and is given by, well, . It is also convenient to note that
These have a fairly complicated differential pattern, but these all follow from the differentials in the Adams spectral sequence for and the Leibniz rule. It is prudent to note that the Adams spectral sequence for has
so for all on the page, and we are left with a sequence of dots separated by . In fact the sequence starts at with and . These again eventually get killed by 's in a manner exactly analogous to the 's.