Equivariant linear isometries and infinite little discs operads via transfer systems

TL;DR

This paper uses transfer systems to relate $G$-equivariant linear isometries and infinite little discs operads, classifying when these operads are homotopically equivalent based on subgroup and representation structures.

Contribution

It introduces a classification of $G$-universes where the linear isometries and little discs operads are homotopically equivalent, utilizing transfer systems to connect topological and algebraic group properties.

Findings

Classified $G$-universes with homotopically equivalent operads.

Provided conditions for compatible transfer system pairs in abelian groups.

Contributed to conjectures on equivariant operad pairs.

Abstract

In this article, we apply the recently developed theory of transfer systems to study the relationship between $G$-equivariant linear isometries and infinite little discs operads, for a finite group $G$. This framework allows us to reduce involved topological problems to discrete problems regarding the subgroup structure and representation theory of the group $G$. Our main result is an example of this: we classify the $G$-universes $\mathcal{U}$ for which the linear isometries operad $\mathcal{L}(\mathcal{U})$ and the infinite little discs operad $\mathcal{D}(\mathcal{U})$ are homotopically equivalent. To achieve this, we use ideas that originate from the work of Balchin-Barnes-Roitzheim on the combinatorics of transfer systems on a total order. Additionally, the use of transfer systems gives us insight into the algebraic structures that arise from equivariant homotopy theory. Compatible pairs of transfer systems provide rules for when multiplicative transfer maps can be paired with additive transfer maps. In the case that the group $G$ is abelian, we provide conditions for when the pair $(\mathcal{L}(\mathcal{U}),\mathcal{D}(\mathcal{U}))$ defines a maximally compatible pair of transfer systems. As a consequence, we contribute to a recent conjecture about equivariant operad pairs.