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?

Solution 1:

(Let's start with the complex case.)

Basically there are two issues here.

1) First of all, Chern classes are stably invariant, i.e. can't distinguish stably isomorphic vector bundles. So the question becomes

Do Chern classes determine an element in the group $\tilde K^0(X)$ of vector bundles on $X$ modulo stable equivalence?

(Example. Since $\pi_4(U(2))=\pi_4(S^1\times S^3)=\mathbb Z/2$ there is a non-trivial 2-bundle on $S^5$. But of course $\tilde K^0(S^5)=0$ so this bundle is stably trivial — in particular, it has zero Chern classes.)

2) This 'stable' variant of the question is (still very nontrivial but) much more tractable (essentially because K-theory is a generalized cohomology theory etc). The Chern character gives an isomorphism $K^0(X)\otimes\mathbb Q\to H^{\text{even}}(X;\mathbb Q)$. So

if $K(X)$ has no torsion, an element of $K(X)$ is completely determined by its Chern classes.

And if $H(X)$ is torsion-free so is $K(X)$ (this follows from AHSS). That's true e.g. for complex Grassmannians and flag manifolds.

(Example. The if $L\in\tilde K^0(\mathbb RP^6)$ is the complexified tautological bundle, $4L$ is a non-trivial element, i.e. a stably non-trivial bundle, with trivial Chern classes.)

Solution 2:

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?

No. For example, the tangent bundle of $S^4$ is stably trivial (because the outward pointing normal gives a trivialization of its normal bundle in $\mathbb{R}^5$; true more generally of all spheres) but not trivial (because its Euler class is nontrivial; true more generally of all even spheres), so its Stiefel-Whitney and Pontrjagin classes vanish but it is not isomorphic to a trivial bundle.