Are Chern classes topological invariants? To be more precise: Given two complex manifolds $M$ and $N$.
Does a homeomophism $f:M\to N$ map Chern classes to Chern classes?
No. Even Chern numbers are not topologically invariant — an example was given by Borel and Hirzebruch in 1959 (Characteristic classes and homogeneous spaces, §24.11).