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the first Chern class $c_1(-):Vect_\mathbb C ^n(-) \rightarrow H^2(-;\mathbb Z)$ is a complete invariant, i.e. yields an isomorphism of groups when evaluated at $X$ with the homotopy type of a CW complex.

I am firstly looking for concrete examples as to why the higher Chern Classes are not complete invariants of higher rank Vector bundles. Also, could suitable subgroups of $Vect_\mathbb C^n(X)$ be chosen such that the restricted Chern classes become complete invariants?

TheHumanHighway
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    Complex vector bundles of rank $k$ over $S^n$ are classified by $\pi_n BU(k) = \pi_{n-1} U(k)$. This is often nonzero in the unstable range with small $k$, even if $n$ is odd and Chern classes can't be zero. However, there's an isomorphism known as the Chern character $K(X) \otimes \Bbb Q \to H^*(X;\Bbb Q)$. –  Dec 24 '16 at 20:09

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