Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Automorphisms of framed operads

Geoffroy Horel, Thomas Willwacher

Abstract

Let $\mathsf P$ be an operad acted upon by a group $G$, and let $\mathsf Q=\mathsf P\rtimes G$ be the corresponding framed operad. We relate the homotopy automorphism groups of $\mathsf P$ and $\mathsf Q$. We apply the result to compute the automorphisms of the framed little disks operad.

Automorphisms of framed operads

Abstract

Let be an operad acted upon by a group , and let be the corresponding framed operad. We relate the homotopy automorphism groups of and . We apply the result to compute the automorphisms of the framed little disks operad.
Paper Structure (16 sections, 20 theorems, 104 equations)

This paper contains 16 sections, 20 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Model categories, functors and adjunctions
  3. Model category structures
  4. Functors and adjunctions
  5. Lambda operads and Reedy model structure
  6. Proof of Theorem \ref{['thm:main']}
  7. Two fibration lemmas
  8. Proof of the first part of Theorem \ref{['thm:main']}
  9. Proof of the second part of Theorem \ref{['thm:main']}
  10. Application to the framed little $n$-disks operads
  11. Proof of Theorem \ref{['thm:main flD top']}
  12. Dg Lie algebras and rational homotopy theory
  13. Proof of Theorem \ref{['thm:main flD Q']}
  14. Proof of Theorem \ref{['thm:main flD profinite']}
  15. A remark about the non-unital case
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\mathop{\mathrm{\mathsfit{P}}}\nolimits$ be (topological or simplicial) operad acted upon by a (topological or simplicial) group $G$ such that $\mathop{\mathrm{\mathsfit{P}}}\nolimits(0)=*$, $\mathop{\mathrm{\mathsfit{P}}}\nolimits(1)$ is contractible, and let $\mathop{\mathrm{\mathsfit{Q}}}\no

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2: Horel HorelProfinite
  • Theorem 1.3: Fresse Frbook
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1: Berger-Moerdijk
  • proof
  • Lemma 2.2
  • proof
  • ...and 25 more