# Differentials in the Adams spectral sequence

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.