On page 283 of Max Karoubi’s book, “K-theory,” he states that for any compact Hausdorff space $X$, the Chern character determines an isomorphism from the group $K^0(X) \otimes Q$ to $H^{even}(X; Q)$, the direct sum of the even-dimensional Cech cohomology groups of X with rational coefficients. In particular, this theorem implies that $K^0(X)$ and $H^{even}(X; Z)$ are isomorphic up to torsion.

Does anyone know of an example (and also a reference, preferably) of a smooth compact manifold $X$ with the property that $K^0(X)$ and $H^{even}(X; Z)$ are not isomorphic?