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.

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.

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.

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.