So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via clutching construction, a bundle over the grassmanian $G(k,n)$ with a preassigned top Chern Class (as dual to some cycle in homology).

To be more precise, I have the answer to the question: "Can you find a bundle of rank $d$ with top Chern class dual to a certain codimension $d$ cycle?"

I also think I can try to tackle with some results the case of the other Chern classes too. It didn't seem too difficult so I was wondering if this result could be found somewhere. After 2/3 months of googling I don't recall stumbling upon it anywhere but the best explanation I have for this is that it is out there, implied by something more complex and I am just still too ignorant about the matter to see it.

So, before 'wasting' more time on it, can anyone crush my dreams and give me a reference to previous works?



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