Let $X$ be a compact differentiable manifold of dimension $m$ or, if you prefer, a smooth complex projective manifold of complex dimension $n=m/2$. 

The Euler characteristic $\chi(X):=\Sigma_{i=0}^{m}(-1)^ib_i(X)$ -the alternating sum of Betti numbers of $X$- is related to the Euler class $e(T_X)$ of the tangent bundle $T_X$ by
$$\chi(X)=\int_Xe(T_X)$$
where $\int_X$ denotes contraction against the fundamental class $[X]$.

Is it possible to write each Betti number $b_i(X)$ as a "characteristic number", i.e. as an integral 
$$b_i(X)\stackrel{\text{?}}{=}\int_X\gamma_i(T_X)$$
of some "characteristic class" $\gamma_i$ of vector bundles? If not, why not? 

This [question][1] is possibly related.


  [1]: https://mathoverflow.net/questions/123913/chern-numbers-via-euler-characteristics