In this post we consider the Adams spectral sequence which computes the stable homotopy groups of spheres at the prime 2.
The classes \(\{h_j\}_{j\ge 0}\) on the 1-line of the second page are called Hopf classes since they are related to the Hopf invariant: Adams proved that Hopf invariant one classes can only exist in dimensions of the form \(2^j – 1\), and there exists such a class if and only if the class \(h_j\) is a permanent cycle (i.e., is in the kernel of all differentials \(d_r\) for \(r \ge 2\)). Further, he proved that \(d_2(h_j) \not= 0\) for all \(j \ge 4\), i.e. this cycles are not permanent. A geometric application of this result is that the only parallelizable spheres are \(S^1\), \(S^3\) and \(S^7\).
On the 2-line we have classes \(\{h_j^2\}_{j\ge 0}\) and Browder came up with the following geometric interpretion for them: Smooth framed manifolds with Kervaire invariant one can only exist in dimensions of the form \(2(2^j – 1)\), and there exists such a manifold if and only if the class \(h_j^2\) is a permanent cycle. It was known from the 70’s and 80’s that the Kervaire classes \(h_j^2\) are permanent cycles for \(0 \le j \le 5\), and Hill-Hopkins-Ravenel proved only a few years ago that \(h_j^2\) is not permanent for all \(j \ge 7\) and thus resolved the Kervaire invariant one problem in high dimensions. The only remaining case is \(h^2_6\) whose fate is currently unknown.
On the 3-line we have classes \(\{h_j^3\}_{j\ge 0}\). Interestingly, there is currently no known geometric interpretation of them. But we do know their fate: In the 70s it was proven that the classes \(h_j^3\) are permanent cycles for \(0 \le j \le 4\), over the last 25 years it was proven that \(d_4(h_j^3) \not= 0\) for \(j = 5,6,7\) (i.e. this cycles are not permanent) and in a recent preprint by Burklund-Xu (arXiv:2302.11869) that \(d_4(h_j^3) \not= 0\) for all \(j \ge 6\). Thus the fate of these classes on the 3-line is now completely settled.
Regarding the case of the Kervaire classes \(h_j^2\) for \(j \ge 6\), as far as I understand the literature it is currently unknown to which page of the spectral sequence these classes survive (it is only known that they eventually die out for \(j \ge 7\)). In this direction, in the mentioned preprint by Burklund-Xu it is proven that \(d_r(h_j^2) = 0\) for \(2 \le r \le 4\) and all \(j\), i.e. all the classes survive to the \(E_5\)-page, and in the remaining mysterious case of \(h_6^2\) we have \(d_r(h_6^2)=0\) for \(r \le 8\), i.e. it survives to the \(E_9\)-page.
One would guess now that investigating higher powers becomes successively harder, but the contrary is actually the case: Adams proved that \(h_j^4 = 0\) for all \(j \ge 1\), and \(h_0^n\) are nonzero permanent cycles.
As a remark, note that there is more stuff on the \(E_2\)-page of the Adams spectral sequence than all the above discussed powers of the Hopf classes.