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$C_2$-Equivariant Orthogonal Calculus

Emel Yavuz

TL;DR

This work develops a $C_2$-equivariant version of orthogonal calculus by introducing a bi-graded index $(p,q)$ and constructing a full model-categorical framework from input functors to stable spectra. The authors define polynomial and homogeneous functors in the equivariant setting, prove two commuting directions of differentiation, and establish a zig-zag of Quillen equivalences connecting the $(p,q)$-homogeneous theory to genuine orthogonal $C_2$-spectra with $O(p,q)$-action. The central result is a Weiss-type classification: every $(p,q)$-homogeneous functor is objectwise equivalent to $V\mapsto \Omega^\infty[(S^{(p,q)V}\wedge\Theta)_{hO(p,q)}]$ for some $\Theta$ in $C_2 Sp^O[O(p,q)]$, and conversely any such form arises from a spectrum. These results extend orthogonal calculus to the equivariant realm, enabling computation of derivatives and classification of layers in contexts with $C_2$-symmetry, with potential applications to equivariant diffeomorphism theories and beyond.

Abstract

In this paper, we construct a version of orthogonal calculus for functors from $C_2$-representations to $C_2$-spaces, where $C_2$ is the cyclic group of order 2. For example, the functor $BO(-)$, that sends a $C_2$-representation to the classifying space of its orthogonal group, which has a $C_2$-action induced by the action on the $C_2$-representation. We obtain a bigraded sequence of approximations to such a functor, and via a zig-zag of Quillen equivalences, we prove that the homotopy fibres of maps between approximations are fully determined by orthogonal spectra with a genuine action of $C_2$ and a naive action of the orthogonal group $O(p,q):=O(\mathbb{R}^{p+qδ})$.

$C_2$-Equivariant Orthogonal Calculus

TL;DR

This work develops a -equivariant version of orthogonal calculus by introducing a bi-graded index and constructing a full model-categorical framework from input functors to stable spectra. The authors define polynomial and homogeneous functors in the equivariant setting, prove two commuting directions of differentiation, and establish a zig-zag of Quillen equivalences connecting the -homogeneous theory to genuine orthogonal -spectra with -action. The central result is a Weiss-type classification: every -homogeneous functor is objectwise equivalent to for some in , and conversely any such form arises from a spectrum. These results extend orthogonal calculus to the equivariant realm, enabling computation of derivatives and classification of layers in contexts with -symmetry, with potential applications to equivariant diffeomorphism theories and beyond.

Abstract

In this paper, we construct a version of orthogonal calculus for functors from -representations to -spaces, where is the cyclic group of order 2. For example, the functor , that sends a -representation to the classifying space of its orthogonal group, which has a -action induced by the action on the -representation. We obtain a bigraded sequence of approximations to such a functor, and via a zig-zag of Quillen equivalences, we prove that the homotopy fibres of maps between approximations are fully determined by orthogonal spectra with a genuine action of and a naive action of the orthogonal group .
Paper Structure (17 sections, 53 theorems, 142 equations)

This paper contains 17 sections, 53 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Equivariant functor categories
  3. The input functors
  4. The intermediate categories
  5. Derivatives
  6. The $(p,q)$-stable model structure
  7. Equivariant Polynomial functors
  8. Polynomial functors
  9. Polynomial approximation
  10. The $(p,q)$-polynomial model structure
  11. Equivariant Homogeneous functors
  12. Homogeneous functors
  13. The $(p,q)$-homogeneous model structure
  14. The equivariant classification theorem
  15. The intermediate category as a category of spectra
  16. ...and 2 more sections

Key Result

Theorem A

Let $p,q\geq 1$. If $F$ is a $(p,q)$-homogeneous functor, then $F$ is objectwise weakly equivalent to where $\Theta_F^{p,q}\in C_2 Sp^{\mathcal{O}}[O(p,q)]$ and $(-)_{hO(p,q)}$ denotes homotopy orbits. Conversely, every functor of the form where $\Theta\in C_2 Sp^{\mathcal{O}}[O(p,q)]$, is $(p,q)$-homogeneous.

Theorems & Definitions (125)

  • Theorem A: Theorem \ref{['weissclassification']}
  • Theorem B: Theorem \ref{['boclassification']} and Theorem \ref{['QEstabletospectra']}
  • Theorem C: Theorem \ref{['thm: DYpq is homog']}
  • Proposition D: Proposition \ref{['cofibseq']}
  • Definition 2.1.1
  • Remark 2.1.2
  • Definition 2.1.3
  • Proposition 2.1.4
  • Definition 2.2.1
  • Theorem 2.2.2: The Equivariant Splitting Theorems
  • ...and 115 more