Comparing orthogonal calculus and calculus with Reality

TL;DR

This paper establishes a connection between calculus with Reality and orthogonal calculus using a $C_2$-fixed points functor, revealing how orthogonal calculus can be recovered up to a shift, similar to real topological K-theory.

Contribution

It introduces a $C_2$-fixed points functor that relates calculus with Reality to orthogonal calculus, providing a new perspective on their relationship.

Findings

Existence of a $C_2$-fixed points functor from calculus with Reality to orthogonal calculus.

Recovery of orthogonal calculus up to a shift via this functor.

Analogy with real topological K-theory and Atiyah's K-theory with Reality.

Abstract

We show that there exists a suitable $C_2$-fixed points functor from calculus with Reality to the orthogonal calculus of Weiss which recovers orthogonal calculus ``up to a shift'' in an analogous way with the recovery of real topological $K$-theory from Atiyah's $K$-theory with Reality via appropriate $C_2$-fixed points.