The $K$-Theory of the Sphere with the Antipodal Involution
Jeffrey L Boersema
TL;DR
The article computes the real K-theory of the sphere S^d endowed with the antipodal involution, obtaining KO_*(\mathfrak S^d) and the united K-theory K^{CR}(\mathfrak S^d) and K^{CRT}(\mathfrak S^d) for all d. It uses the unitary picture of real K-theory to provide explicit generators in low dimensions (d≤4) and presents a Clifford-based algorithm to generate the key KO_{d+2}(\mathfrak S^d) classes for all d, with a clear dependence on d mod 8. A central structural result is that for d≥2, KO_*(\mathfrak S^d) fits the decomposition KO_*(\mathbb{R}) ⊕ Σ^{-d-2} KO_*(\mathbb{R}) and K^{CRT}(\mathfrak S^d) is a free CRT-module, enabling concrete calculations via the CRT framework. The work extends complex K-theory methods to the real setting, provides practical unitary representatives for physical-model markers, and includes an appendix establishing equivalence between alternative unitary pictures for KO_{-1} and KO_3, ensuring robustness of the methods across formulations.
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 $d$.