Every Endomorphism of the Framed Little Disk Operad is an Automorphism

TL;DR

This paper extends the result that all endomorphisms of the little disk operad are automorphisms to the framed version, using Lie group classification, and explores similar properties for related operads.

Contribution

It proves that every endomorphism of the framed little disk operad is an automorphism, generalizing previous results and analyzing related operads.

Findings

All endomorphisms of the framed little disk operad are automorphisms.

The property holds for some semidirect products of groups with the little disk operad.

The classification of self maps of simple Lie groups is used in the proof.

Abstract

In a recent paper, Horel-Krannich-Kupers proved that all endomorphisms of the little $d$-disk operad are automorphisms. In this paper we show that this is also true for the framed little $d$-disk operad by using the classification of self maps of simple Lie groups. We also examine whether this property holds for the swiss cheese operad and prove that it holds for some other semidirect products of a group with a little disk operad.