Quotients of even rings

TL;DR

This paper proves that for certain even-degree rings, quotient constructions by arbitrary sequences of elements can be given an $bE_1$-algebra structure, removing the regular sequence assumption common in previous work.

Contribution

It establishes that quotients of even-degree $bE_2$-rings by arbitrary sequences admit an $bE_1$-algebra structure, broadening the class of quotients in algebraic topology.

Findings

Quotients of even $bE_2$-rings by arbitrary sequences have an $bE_1$-structure.

Removes the regular sequence assumption in quotient constructions.

Extends the understanding of algebraic structures in homotopy theory.

Abstract

We prove that if $R$ is an $\mathbb{E}_2$-ring with homotopy concentrated in even degrees, and $\{x_j\}$ is any sequence of elements in $\pi_{2*}(R)$, then $R/(x_1,x_2,\cdots)$ admits the structure of an $\mathbb{E}_1$-$R$-algebra. This removes an assumption, common in the literature, that $\{x_j\}$ be a regular sequence.