TITLE: Betti numbers as characteristic numbers?
QUESTION [9 upvotes]: 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 is possibly related.

REPLY [17 votes]: No. The Stiefel-Whitney and Pontryagin numbers of a closed oriented manifold are cobordism invariants, but the Betti numbers are not. 
More explicitly, all closed oriented $3$-manifolds are frameable (and null cobordant), so all of their characteristic numbers vanish, but e.g. the $3$-torus and $3$-sphere have different Betti numbers.