On the algebras over equivariant little disks

TL;DR

This paper explores the algebraic structures over equivariant little disks operads for various group representations, including non-universe cases, with detailed analysis for the cyclic group of order two and applications to homology computations.

Contribution

It characterizes the algebraic structures over equivariant little disks operads for diverse representations and provides explicit descriptions for the sign representation case, including homology implications.

Findings

Describes algebraic structures over equivariant little disks operads for various G-representations.

Provides a detailed analysis for G=C2 and the sign representation.

Determines the homology of the signed James construction.

Abstract

We describe the structure present in algebras over the little disks operads for various representations of a finite group $G$, including those that are not necessarily universe or that do not contain trivial summands. We then spell out in more detail what happens for $G=C_{2}$, describing the structure on algebras over the little disks operad for the sign representation. Here we can also describe the resulting structure in Bredon homology. Finally, we produce a stable splitting of coinduced spaces analogous to the stable splitting of the product, and we use this to determine the homology of the signed James construction.