On the E2‐term of the bo‐Adams spectral sequence

Abstract

The E 1 -term of the (2-local) bo-based Adams spectral sequence for the sphere spectrum decomposes into a direct sum of a v 1 -periodic part, and a v 1 -torsion part. Lellmann and Mahowald completely computed the d 1 -differential on the v 1 -periodic part, and the corresponding contribution to the E 2 -term. The v 1 -torsion part is harder to handle, but with the aid of a computer it was computed through the 20-stem by Davis. Such computer computations are limited by the exponential growth of v 1 -torsion in the E 1 -term. In this paper, we introduce a new method for computing the contribution of the v 1 -torsion part to the E 2 -term, whose input is the cohomology of the Steenrod algebra. We demonstrate the efficacy of our technique by computing the bo-Adams spectral sequence beyond the 40-stem.