# On the E2-term of the bo-Adams spectral sequence

## TL;DR

This paper introduces a new computational method for analyzing the v_1-torsion part of the bo-Adams spectral sequence, enabling calculations beyond the 40-stem that surpass previous computer limitations.

## Contribution

A novel approach utilizing Steenrod algebra cohomology to compute the v_1-torsion contribution to the E_2-term in the bo-Adams spectral sequence.

## Key findings

- Successfully computed the bo-Adams spectral sequence beyond the 40-stem.
- Demonstrated the method's effectiveness in handling complex torsion components.
- Extended understanding of the spectral sequence's structure in higher stems.

## 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.