The Fiber of $Sq^n$

TL;DR

This paper proves that certain Adams spectral sequences related to the fiber of a map between GEMs collapse at E_3, simplifying the calculation of homotopy classes.

Contribution

It demonstrates the collapse of specific Adams spectral sequences at E_3 without needing to compute the fiber's cohomology or E_2 terms.

Findings

Spectral sequences collapse at E_3

E_3 equals E_infinity in these cases

Simplifies homotopy class calculations

Abstract

A colleague asked about the Adams filtrations of the homotopy classes in the homotopy of the fiber of a particular map between GEMs. The theorem proved in arXiv:2105.02601v3 [math.AT] proves to be effective in answering this (Theorem 4.4). We show that this and some related Adams spectral sequences all collapse at $E_3$ and we determine the value of $E_3 = E_\infty$. Notably, we do not need to determine the cohomology of the fiber or the $E_2$ term of the Adams spectral sequence to do this.