Realizing Transfer Systems as Suboperads of Coinduced Operads

TL;DR

This paper introduces a method to identify specific suboperads within coinduced operads that realize arbitrary transfer systems, applicable to various well-known operads, and constructs an intersection _-infinity operad for this purpose.

Contribution

It provides an explicit construction for suboperads of coinduced operads that realize any transfer system, expanding the understanding of operad structures in algebraic topology.

Findings

Method works for intersection operads including little k-cube, Steiner, and linear isometries operads.

Constructs an intersection _-infinity operad realizing any transfer system.

Enables explicit realization of transfer systems within well-known operads.

Abstract

In this paper, we present an explicit method to identify equivariant suboperads of coinduced operads that contain only fixed points associated to any desired transfer system. Our method works for a class of operads that we call intersection operads, which includes many familiar operads of interest, including the little $k$-cube operads, the Steiner operad, and the linear isometries operad. As an application, we also construct an intersection $\mathbb{E}_\infty$-operad that, when applying our construction, will produce a $\mathbb{N}_\infty$-operad realizing an arbitrary transfer system.