$C_2$-Equivariant Orthogonal Calculus

TL;DR

This thesis develops a $C_2$-equivariant orthogonal calculus, constructing bigraded polynomial approximations for functors from $C_2$-representations to $C_2$-spaces, and classifying homogeneous functors via equivariant spectra.

Contribution

It introduces a new $C_2$-equivariant orthogonal calculus framework with bigraded approximations and classifies homogeneous functors using equivariant spectra.

Findings

Constructed strongly $(p,q)$-polynomial approximations $T_{p,q}F$.

Proved classification of $(p,q)$-homogeneous functors via equivariant spectra.

Established equivalences relating functors to spectra with $C_2$ and $O(p,q)$ actions.

Abstract

In this thesis, we construct a new version of orthogonal calculus for functors $F$ from $C_2$-representations to $C_2$-spaces, where $C_2$ is the cyclic group of order 2. For example, the functor $BO(-)$, which sends a $C_2$-representation $V$ to the classifying space of its orthogonal group $BO(V)$. We obtain a bigraded sequence of approximations to $F$, called the strongly $(p,q)$-polynomial approximations $T_{p,q}F$. The bigrading arises from the bigrading on $C_2$-representations. The homotopy fibre $D_{p,q}F$ of the map from $T_{p+1,q}T_{p,q+1}F$ to $T_{p,q}F$ is such that the approximation $T_{p+1,q}T_{p,q+1}D_{p,q}F$ is equivalent to the functor $D_{p,q}F$ itself and the approximation $T_{p,q}D_{p,q}F$ is trivial. A functor with these properties is called $(p,q)$-homogeneous. Via a zig-zag of Quillen equivalences, we prove that $(p,q)$-homogeneous functors are fully determined by orthogonal spectra with a genuine action of $C_2$ and a naive action of the orthogonal group $O(p,q)$.