The Adams differentials on the classes $h_{j}^{3}$

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 $h_{j}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> , 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 $h_{j}^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mi>j</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> support non-trivial differentials for $j \geq 7$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>j</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>7</mml:mn> </mml:math> , resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes $h_{j}^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mi>j</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> for $j \geq 7$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>j</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>7</mml:mn> </mml:math> and the fate of $h_{6}^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> remains unknown. In this paper, in Adams filtration 3, we prove an infinite family of non-trivial $d_{4}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> -differentials on the classes $h_{j}^{3}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mi>j</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:math> for $j \geq 6$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>j</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>6</mml:mn> </mml:math> , confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory—ℂ-motivic stable homotopy theory and $\mathbb{F}_{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -synthetic homotopy theory—both in an essential way. Along the way, we also show that $h_{j}^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mi>j</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> survives to the Adams $E_{5}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:math> -page and that $h_{6}^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> survives to the Adams $E_{9}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>9</mml:mn> </mml:msub> </mml:math> -page.