C2-equivariant orthogonal calculus

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).

Date of Award	Dec 2024
Original language	English
Awarding Institution	Queen's University Belfast
Sponsors	Engineering and Physical Sciences Research Council
Supervisor	David Barnes (Supervisor) & Lisa McFetridge (Supervisor)