Odd topological K-theory ($K^1(X)=[X,U_{\mathcal{K}}]$)

For compact metric $X$, odd topological K-theory is the homotopy set of norm-continuous maps into unitaries equal to the identity modulo compact operators on a fixed separable infinite-dimensional Hilbert space. Uniform finite-rank approximation over $X$ and over homotopies gives the equivalent stable model ${\mathrm{colim}}_N[X,G{L}_N(\mathbb{C})]$. Block sum gives an abelian group: a finite-dimensional unitary path swaps blocks, and a two-block rotation contracts $\mathrm{diag}(u,u^{-1})$ to the identity. On the circle the determinant winding identifies $K^1(S^1)$ with $\mathbb{Z}$; Bott periodicity gives $K^1(S^n)=\mathbb{Z}$ in odd dimensions and zero in even dimensions.