The $K$-Theory of the Sphere with the Antipodal Involution

TL;DR

This paper thoroughly investigates the real K-theory of spheres with antipodal involution, providing explicit calculations, generators, and a method for all dimensions, advancing understanding of equivariant K-theory.

Contribution

It offers comprehensive calculations of real K-theory for spheres with antipodal symmetry, including explicit generators and a general construction method for all dimensions.

Findings

Calculated algebraic structure of real K-theory for all dimensions

Explicit unitaries for generators in dimensions up to 4

Provided a general recipe for generators in all dimensions

Abstract

This is a thorough investigation on the real $K$-theory of the sphere $S^d$ associated with the antipodal involution. We calculate the algebraic structure of real $K$-theory and united $K$-theory for all $d$, we write down explicit unitaries representing the generators of all the non-trivial $K$-theory groups for $d \leq 4$, and we describe a recipe for generating such unitaries for all $d$.