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I asked the same question on mathstackexchange but didn't get any response so I guess I should ask here. If you think it is a duplicate and it is not appropriate to post it here, I will delete this question.

I am reading MacPherson's paper "Chern Classes for singular varieties".

Proposition1 : There is a unique covariant functor from compact complex algebraic variety to abelian groups whose value on a variety is the group of constructible functions from that variety to the integers and whose value $f_*$ on a map $f$ satisifies $$f_*(1_W)(p)=\chi(f^{-1}(p)\cap W)$$, where $1_W$ is the function takes 1 on $W$ and zero otherwise, and $\chi$ denotes the toplogical Euler characteristic.

In the proof, he mentioned stratification theory but I am not very familiar with that. I don't see why $f_*(1_w)$ is constructible function on $Y$. Is there anyone who can explain this to me? Thank you so much.

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  • $\begingroup$ The fact one needs to use is that given any morphism $g:X \to Y$ of complex algebraic varieties, there is a nonempty open set $U \subset Y$ so that $g|_{g^{-1}(U)}: g^{-1}(U) \to U$ is a (topological) fibre bundle. $\endgroup$
    – naf
    Commented Feb 9, 2016 at 6:15
  • $\begingroup$ @ulrich where can I find this fact? Thank you so much. $\endgroup$
    – Ben
    Commented Feb 9, 2016 at 22:11
  • $\begingroup$ A reference is the Corollary on the last page of the paper "Stratifications de Whitney et Theoremes de Bertini-Sard" by Verdier, Inventiones 36. $\endgroup$
    – naf
    Commented Feb 10, 2016 at 4:47

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