= Solution
Fix a separable infinite-dimensional complex <Hilbert space> $\mathcal H$ and let
$$
U_{\mathcal K}=\{u\in B(\mathcal H):u\text{ unitary},\ u-I\text{ compact}\}
$$
with its operator-norm topology. The <odd topological K-theory> group of a compact metric space is defined by continuous-map <homotopy> classes
$$
\boxed{K^1(X)=[X,U_{\mathcal K}].}
$$
<Homotopies> use the same norm topology. Its finite-dimensional stable realization and group operation are established in the following parts, so this definition does not yet assume that <homotopy> classes form a group.
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