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I wanted to check whether the following characterization of odd complex topological $K$-theory is correct (reposted from Math.SE). Let $X$ be a compact Hausdorff space. Then $K^{-1}(X)$ can be defined ...

Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how ...