TITLE: Chern numbers via Euler characteristics?
QUESTION [12 upvotes]: Let $X$ be a space good enough to have a fundamental class, and $E$ a complex vector bundle on $X$.  Let $P$ be some polynomial expression, and say I want to evaluate $P(c_i(E)) \cap [X]$. 

Is there some naturally associated space, $\mathbf{P}(X, E)$ such that (e.g.) the topological Euler characteristic $\chi(\mathbf{P}(X, E)) = P(c_i(E)) \cap [X]$?

REPLY [2 votes]: Let $B = Gr_{\dim E}(\infty)$ denote the classifying space for $\dim E$-bundles. Your $P(c_i(E))$ defines a class on $B$, which (if $P$ has integer coefficients, which you better've meant it to!) can pretty obviously be represented by an actual cycle, $\mathcal P \subseteq B$.
Assume the classifying map $c : X \to B$ is transverse to $\mathcal P$. Then $c^{-1}(\mathcal P)$ is the space you're looking for. Too bad it's finite!
Not quite. I want $c$ to be not just transverse to $\mathcal P$, but to meet it positively (unless you're okay with $\chi$ of a "negative point" being $-1$). Something like, $X$ should be algebraic, and the classifying map should be algebraic, some condition like $E$ being globally generated.