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Equivariant Framed Little Disk Operads are Additive

Ben Szczesny

Abstract

In this paper, we generalize the Dunn-Brinkmeier~additivity theorem, which establishes a weak equivalence $\mathcal{C}_n \otimes \mathcal{C}_m \simeq \mathcal{C}_{n+m}$ for the little cubes operad $\mathcal{C}_n$. We introduce equivariant framed little disk operads, a new class of operads that simultaneously generalize the framed little disk operads and the little $V$-disk operads associated with a $G$-representation $V$. We prove that these operads satisfy an analogous additivity property, extending the classical theorem to settings involving group actions and framings.

Equivariant Framed Little Disk Operads are Additive

Abstract

In this paper, we generalize the Dunn-Brinkmeier~additivity theorem, which establishes a weak equivalence for the little cubes operad . We introduce equivariant framed little disk operads, a new class of operads that simultaneously generalize the framed little disk operads and the little -disk operads associated with a -representation . We prove that these operads satisfy an analogous additivity property, extending the classical theorem to settings involving group actions and framings.

Paper Structure

This paper contains 21 sections, 33 theorems, 223 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Outline of the Paper
  3. Acknowledgments
  4. Conventions
  5. General Conventions
  6. Operads Indexed on Finite Ordinals
  7. Components of Reduced Operads
  8. Boardman-Vogt Tensor and Trees in Superposition
  9. The Boardman-Vogt Tensor of Operads
  10. Trees in Superposition
  11. Representatives in the Tensor
  12. Equivariant Framed Little Disks
  13. Comparisons Between Framed Disk Operads
  14. Divisibility in Operads
  15. Additivity of Equivariant Framed Little Disks
  16. ...and 6 more sections

Key Result

Theorem 1.1

Given $G$-representations $V$ and $W$, and dilation representations $\rho: \mathcal{G}\to \mathop{\mathrm{GL}}\nolimits(V)$, and $\psi: \mathcal{H}\to \mathop{\mathrm{GL}}\nolimits(W)$, there is a weak equivalence of $G$-operads

Figures (4)

  • Figure 7.1: The extreme case
  • Figure 7.2: An element of $\mathop{\mathrm{\mathscr{K}}}\nolimits^\rho(B_V)$.
  • Figure 7.3: Example for indexing partitions
  • Figure 7.4: Example of constructed elements

Theorems & Definitions (91)

  • Theorem 1.1
  • Definition 2.3
  • Remark 2.4
  • Example 2.9
  • Definition 2.10
  • Lemma 2.11
  • Definition 2.12
  • Theorem 2.13
  • Theorem 3.1
  • Definition 3.2
  • ...and 81 more