A comparison of spectral sequences computing unstable homotopy groups of $p$-complete, nilpotent spaces

TL;DR

This paper compares two spectral sequences used to compute unstable homotopy groups of p-complete, nilpotent spaces, establishing an isomorphism between them through an adjunction, thus linking different computational approaches.

Contribution

It demonstrates that the unit of a specific adjunction induces an isomorphism between the unstable mod p Adams spectral sequence and the Goerss–Hopkins spectral sequence.

Findings

The spectral sequences are isomorphic via the adjunction unit.

The comparison provides a unified understanding of unstable homotopy computations.

The results apply to p-complete, nilpotent spaces for odd primes.

Abstract

The focus of this paper is the comparison of two unstable homotopy spectral sequences-- the unstable mod p Adams spectral sequence that computes the unstable homotopy of a p-complete space, and the Goerss--Hopkins spectral sequence, which computes the unstable homotopy of the space of E-infinity maps between Hk-algebras, where k is the algebraic closure of the field with p elements and p is an odd prime. Using an adjunction between p-complete nilpotent spaces and a subset of Hk-algebras, this paper shows that the unit of this adjunction provides an isomorphism between these spectral sequences.