Year: 2024
Type: article
Abstract: In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams (Ann. Math. (2) 72:20–104 1960) computed the differentials on the classes $h_{j}$ h j , resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill–Hopkins–Ravenel (Ann. Math. (2) 184(1):1–262 2016) proved that the classes $h_{j}^{2}$ h j 2 support non-trivial differentials for $j \geq 7$ j ≥ 7 , resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes $h_{j}^{2}$ h j 2 for $j \geq 7$ j ≥ 7 and the fate of $h_{6}^{2}$ h 6 2 remains unknown. In this paper, in Adams filtration 3, we prove an infinite family of non-trivial $d_{4}$ d 4 -differentials on the classes $h_{j}^{3}$ h j 3 for $j \geq 6$ j ≥ 6 , confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory—ℂ-motivic stable homotopy theory and $\mathbb{F}_{2}$ F 2 -synthetic homotopy theory—both in an essential way. Along the way, we also show that $h_{j}^{2}$ h j 2 survives to the Adams $E_{5}$ E 5 -page and that $h_{6}^{2}$ h 6 2 survives to the Adams $E_{9}$ E 9 -page.
Source: Inventiones mathematicae
Authors: Robert Burklund, Zhouli Xu
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4404858652