Motivic Stable Homotopy and the Stable 51 and 52 Stems

TL;DR

This paper uses motivic stable homotopy theory to determine a crucial differential in the Adams spectral sequence, leading to the first accurate calculations of the stable 51 and 52 stems, showcasing motivic methods' power.

Contribution

It introduces a novel application of the motivic Adams spectral sequence to resolve an undetermined differential in classical stable homotopy groups.

Findings

Calculated the differential d_2(D_1)=h_0^2h_3g_2 in the 51-stem

Achieved the first correct computation of the 51 and 52 stems

Demonstrated the effectiveness of motivic methods in classical homotopy theory

Abstract

We establish a differential $d_2(D_1)=h_0^2h_3g_2$ in the $51$-stem of the Adams spectral sequence at the prime $2$, which gives the first correct calculation of the stable 51 and 52 stems. This differential is remarkable since we know of no way to prove it without recourse to the motivic Adams spectral sequence. It is the last undetermined differential in the range of the first author's detailed calculations of the $n$-stems for $n<60$ [6]. This note advertises the use of the motivic Adams spectral sequence to obtain information about classical stable homotopy groups.