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Let $E$ be an oriented vector bundle with an even rank $n$ over a smooth oriented $n$-dimensional manifold $M$, $e(E)$ denotes the corresponding Euler class, then the Euler number is defined by $\int_Me(E)$. If $E=TM$ with $M$ closed and oriented, then the Euler number coincides with the Euler characteristic $\chi(M)$. When $M$ is compact, the Euler number and the Euler characteristic are trivially finite. When $M$ has a finite good cover, the finiteness of $\chi(M)$ follows from the isomorphsim between the de Rham cohomology and the Cech cohomology.

QUESTION Is $\int_Me(E)$ finite for every smooth oriented manifold $M$?

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Then the Euler class vanishes since the corresponding cohomology group vanishes. – Acky Nov 17 '11 at 16:14
Sorry, it's my mistake. – Acky Nov 17 '11 at 16:21

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