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

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 $d$.