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Equivariant formality of the little disks operad

Pedro Boavida de Brito, Joana Cirici, Geoffroy Horel

Abstract

The little $n$-disks operad is $SO(n)$ and $O(n)$-equivariantly formal over the rationals. Equivalently, the oriented and unoriented framed little disks operads are rationally formal as $\infty$-operads.

Equivariant formality of the little disks operad

Abstract

The little -disks operad is and -equivariantly formal over the rationals. Equivalently, the oriented and unoriented framed little disks operads are rationally formal as -operads.
Paper Structure (16 sections, 21 theorems, 86 equations)

This paper contains 16 sections, 21 theorems, 86 equations.

Table of Contents

  1. Equivariant homotopy theory
  2. Borel construction and rational equivariant homotopy theory
  3. Leray-Serre spectral sequence for the Borel construction
  4. The orthogonal group
  5. Rational equivariant homotopy theory for the orthogonal group
  6. Homotopy orbits of the little disks $\infty$-operad
  7. Little disks and some special automorphisms
  8. A special automorphism of $\mathcal{D}_2$
  9. A special automorphism of $\mathcal{D}_n$
  10. Purity and filtered formality
  11. Prelude
  12. Proof of the theorem
  13. Equivariant cohomology ring of configuration spaces
  14. Examples in the case of two points
  15. General case in even dimensions
  16. ...and 1 more sections

Key Result

Theorem 1

Let $n\geq 3$, let $G$ be $O(n)$ or $SO(n)$. The little $n$-disks operad $\mathcal{D}_n$ is $G$-equivariantly formal.

Theorems & Definitions (49)

  • Theorem
  • Theorem
  • Theorem
  • Theorem 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 1.7
  • ...and 39 more