$E_k$-pushouts and $E_{k+1}$-tensors

TL;DR

This paper establishes a general framework connecting pushouts of $E_k$-algebras to relative tensors over $E_{k+1}$-algebras, with applications to algebraic topology and homotopy theory, including Steenrod algebra quotients and spectral sequences.

Contribution

It introduces a unifying theorem relating $E_k$-pushouts to $E_{k+1}$-tensors, extending known results and constructing new algebraic structures in stable homotopy theory.

Findings

Realized quotients of the dual Steenrod algebra as associative algebras over $HF_p \wedge HF_p$

Constructed a filtered $E_2$-algebra structure on the sphere spectrum

Spectral sequence for stable homotopy groups with $E_1$-term related to May spectral sequence

Abstract

We prove a general result that relates certain pushouts of $E_k$-algebras to relative tensors over $E_{k+1}$-algebras. Specializations include a number of established results on classifying spaces, resolutions of modules, and (co)homology theories for ring spectra. The main results apply when the category in question has centralizers. Among our applications, we show that certain quotients of the dual Steenrod algebra are realized as associative algebras over $HF_p \wedge HF_p$ by attaching single $E_1$-algebra relation, generalizing previous work at the prime $2$. We also construct a filtered $E_2$-algebra structure on the sphere spectrum, and the resulting spectral sequence for the stable homotopy groups of spheres has $E_1$-term isomorphic to a regrading of the $E_1$-term of the May spectral sequence.