= Odd topological K-theory
{title2=$K^1(X)=[X,U_{\mathcal K}]$}
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 $\operatorname{colim}_N[X,GL_N(\mathbb C)]$. Block sum gives an abelian group: a finite-dimensional unitary path swaps blocks, and a two-block rotation contracts $\operatorname{diag}(u,u^{-1})$ 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.
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