# When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$.

What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on this space is determined by its characteristic classes (Chern or Stiefel-Whitney and Pontrjagin classes)?

For example, in the case of complex line bundles it is enough to have $H^1(X,\mathbb C)=0.$ This comes from the exponential sequence which doesn't exist in higher dimension.

Perhaps for $k>1$ this question is difficult to answer in general, so I'll be more specific.

Is it true for compact simply-connected oriented smooth 4-manifolds that a real 4-dimensional vector bundle is determined by its Stiefel-Whitney and Pontrjagin classes? This would give a partial answer to my yesterday's question (Tangent bundles of exotic manifolds) since in this case characteristic classes of tangent bundle give a purely topological information (signature, Euler number and diagonal of the intersection form mod $2$).

It seems well known that over complex projective space (and even over Grassmannian) in each dimension a complex vector bundle is determined by its total Chern class. Is it true in the real case (or in the mixed case of complex bundles over real Grassmannians or vice versa)? Can we say something about another algebraic varieties?

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I found that the question about 4-manifolds is almost covered by Dold-Whitney Theorem: "If two oriented 4-plane bundles over an oriented 4-manifold have the same second Stiefel-Whitney class $w_2$, Pontryagin class $p_1$ and Euler class $e$, then they must be isomorphic". –  Misha Aug 19 '13 at 10:36