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Every Endomorphism of the Framed Little Disk Operad is an Automorphism

Alice Rolf

TL;DR

This work proves that every endomorphism of the framed little $d$-disk operad $E_d^{\mathrm{SO}(d)}$ is an automorphism, extending the unframed result for $E_d$ to the framed setting. The authors leverage a tangential-structure viewpoint to reduce endomorphisms to maps $BG\to BG$ compatible with the $G$-action, and then classify self-maps of $BG$ for simple compact Lie groups using unstable Adams operations and Weyl-group data, with detailed analysis for $\mathrm{SO}(2n+1)$ and $\mathrm{SO}(2n)$. They further extend the automorphism results to variants such as $E_d^{\mathrm{O}(d)}$, $E_{2n+1}^{\mathrm{SO}(2n)}$, and the (framed) Swiss Cheese operad, establishing that color-preserving endomorphisms are automorphisms and delineating when color-nonpreserving maps can exist (notably ruling out maps sending $D^d$ to $HD^d$). Collectively, the results elucidate the automorphism structure of framed and related operads and have implications for the homotopy theory of mapping spaces and higher algebraic structures in topology, connecting operad theory with Lie-theoretic classification data via $BG$-maps and Adams operations.

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.

Every Endomorphism of the Framed Little Disk Operad is an Automorphism

TL;DR

This work proves that every endomorphism of the framed little -disk operad is an automorphism, extending the unframed result for to the framed setting. The authors leverage a tangential-structure viewpoint to reduce endomorphisms to maps compatible with the -action, and then classify self-maps of for simple compact Lie groups using unstable Adams operations and Weyl-group data, with detailed analysis for and . They further extend the automorphism results to variants such as , , and the (framed) Swiss Cheese operad, establishing that color-preserving endomorphisms are automorphisms and delineating when color-nonpreserving maps can exist (notably ruling out maps sending to ). Collectively, the results elucidate the automorphism structure of framed and related operads and have implications for the homotopy theory of mapping spaces and higher algebraic structures in topology, connecting operad theory with Lie-theoretic classification data via -maps and Adams operations.

Abstract

In a recent paper, Horel-Krannich-Kupers proved that all endomorphisms of the little -disk operad are automorphisms. In this paper we show that this is also true for the framed little -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.

Paper Structure

This paper contains 15 sections, 18 theorems, 33 equations, 3 figures.

Table of Contents

  1. Set-Up
  2. Definitions
  3. Endomorphisms of the $E_2$ operad
  4. Compact Lie groups and maps on their classifying spaces
  5. Compact Lie groups
  6. Maps on classifying spaces of Lie groups
  7. Proof of the main theorem
  8. Self Maps of $B\mathrm{SO}(2n+1)$
  9. Self Maps of $B\mathrm{SO}(2n)$
  10. Variants
  11. $\mathrm{O}(n)$ acting on $E_n$
  12. $\mathrm{SO}(2n)$ acting on $E_{2n+1}$
  13. Endomorphisms of the (framed) Swiss Cheese Operad
  14. Endomorphisms of the swiss cheese operad preserving colors
  15. Endomorphism of the swiss cheese operad not preserving colors

Key Result

Theorem 1

All endomorphisms of ${E_d}^{\mathrm{SO}(d)}$ are automorphisms.

Figures (3)

  • Figure 1: An element in $E_2^{SO(2)}(3)$
  • Figure 2: Dynkin diagrams of $\mathfrak{so}(2n)$ and $\mathfrak{so}(8)$. Their construction is described in fulton2013representation
  • Figure 3: An element in $\mathcal{SC}(3,3)$

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Proposition 1.6
  • proof
  • ...and 34 more