The usual construction for finding torsion elements on complex $K$ theory is using flat vector bundles. So is it still possible to find a simply connected compact space with a nonzero torsion in its $K$ theory. Such an example would be given by a map $f:X\to BU$ which is zero on real cohomology but is not null homotopic. By Bott periodicity theorem, it follows that such a map would be zero on homotopy groups (because $\pi^s\_\*(Y)\otimes \mathbb{R}=H\_\*(Y, \mathbb{R})=H^\*(Y,\mathbb{R})$ and the homotopy groups of $BU$ are free abelian). I don't know if such a map could exist or not. I am most interested in the case of a closed simply connected manifold but an example in the general case would be welcome.



Torsion In $K$ theory on simply connected manifolds
Let $X$ be a $2$-connected closed $7$-manifold with $H\_3(X) = H^4(X) = \mathbb{Z}/2$. Then $\tilde K(X) = \widetilde{KU}^0(X)$ equals $\mathbb{Z}/2$. This follows from the Atiyah--Hirzebruch spectral sequence.


If you do not require $X$ to be a manifold, then the $4$-skeleton of the manifold above, i.e., $S^3 \cup\_2 e^4 = \Sigma^2 \mathbb{R} P^2$, is a slightly smaller example.


