For my graduate (master) thesis I am studying the theory of Chern Classes. As a possible personal development the only sensible idea I have so far, and which I frankly think is impossible, is to work on the inverse problem, i.e., given a class in cohomology, is there a vector bundle which has that class as one of its Chern classes? The following question is to understand what it is already known about this and if maybe I might focus my attention on a small sub-problem and if it is an idea worth telling my supervisor at all or just forget about it.


1. Has this problem been analysed before? Is it a sensible matter to tackle?
2. I know how to find Chern Classes by taking conjugate invariant symmetric polynomials of the curvature matrix of a connection on the manifold (so de Rham Cohomology). Is this approach better, worse, equivalent to studying line divisors, picard groups and related? (Is there any other approach to finding Chern classes?)


**EDIT**


Ok, the problem seems rather involved with fundamental questions. Can you suggest one or more sub-problem which I might be able to work on in 3 months? Maybe even pre-existing results over which I can elaborate which explicit computations, examples, counter-examples, generalizations...



Inverse problem of Chern Classes
For $c\_1$ the problem is solved. $\newcommand{\bZ}{\mathbb{Z}}$ For any smooth manifold and any $c\in H^2(M,\bZ)$ there exists a smooth complex line bundle $L\to M$ such that $c\_1(L)=c$.


By results of Thom, for any oriented manifold $M$, any $\alpha\in H\_{n-4}(M,\bZ)$ is represented by an oriented submanifold.


On the other hand, for any $n\geq 7$, there exists an $n$-dimensional oriented manifold $M$ and a homology class $\alpha\in H\_{n-4}(M,\bZ)$ such that the normal bundle of any submanifold representing $\alpha$ does not admit a $spin^c$-structure; see Theorem 3, page 9 of [this paper](http://arxiv.org/pdf/math/0011178.pdf).


If $\alpha^\dagger\in H^4(M,\bZ)$ denotes the Poincare dual of such an $\alpha$, then there exist no rank 2 complex vector bundle $E\to M$ such that $c\_2(E)=\alpha^\dagger$. 


If such a bundle existed, then the zero set of a generic section of $E$ will be an oriented submanifold $S$ of $M$ representing $\alpha$. The normal bundle of $S$ in $M$ is isomorphic to $E|\_S$. In particular it admits $spin^c$ structures because it admits an almost complex structure.


**Edit 1.** A rather deep divisibility theorem shows that if $n\geq 3$ and $E\to S^{2n}$ is a complex vector bundle, then $c\_n(E)\in H^{2n}(S^{2n},\bZ)$ is divisible by $(n-1)!$. 


