Real and quaternionic bundle K-groups

For a compact Hausdorff space $X$, use the direct-sum monoid of finite-rank bundles whose rank is locally constant and finite-valued. Equivalently, such a bundle is specified by a finite clopen decomposition of $X$ into rank strata and a fixed-rank bundle on each stratum. For real bundles the fixed-rank pieces use the charts of Real and complex topological vector bundles; a right-$\mathbb{H}$ bundle uses local charts with right-$\mathbb{H}$-linear transition maps. Let $K{O}^0(X)$ be the group completion of the real-bundle monoid and $KS{p}^0(X)$ the group completion of the right-$\mathbb{H}$-bundle monoid. This permits ranks to vary between connected components. In particular these definitions apply to finite CW complexes. A group-completion element is $[E]-[F]$; equality means that a bundle $G$ exists with $E\oplus F^{\mathrm{\prime }}\oplus G\cong E^{\mathrm{\prime }}\oplus F\oplus G$ for the two differences concerned.

Pullback defines contravariant maps. Real tensor product makes $K{O}^0(X)$ a commutative ring with unit $[\mathbb{R}]$. Tensoring a real bundle with a right-$\mathbb{H}$ bundle over $\mathbb{R}$ gives a $K{O}^0(X)$-module structure on $KS{p}^0(X)$. No internal quaternionic tensor product is asserted.