Equivariant Framed Little Disk Operads are Additive

TL;DR

This paper extends the classical additivity theorem for little disks operads to equivariant framed versions, incorporating group actions and framings, thus broadening the operadic framework in algebraic topology.

Contribution

It introduces equivariant framed little disk operads and proves they satisfy an additivity property similar to the classical theorem.

Findings

Generalization of the additivity theorem to equivariant framed operads

Introduction of a new class of operads combining framing and group actions

Proof of weak equivalence for the new operads similar to classical case

Abstract

In this paper, we generalize the Dunn-Brinkmeier~additivity theorem, which establishes a weak equivalence $\mathcal{C}_n \otimes \mathcal{C}_m \simeq \mathcal{C}_{n+m}$ for the little cubes operad $\mathcal{C}_n$. We introduce equivariant framed little disk operads, a new class of operads that simultaneously generalize the framed little disk operads and the little $V$-disk operads associated with a $G$-representation $V$. We prove that these operads satisfy an analogous additivity property, extending the classical theorem to settings involving group actions and framings.