cohomology fiber bundles I will be infinitely grateful to the one who could give a thorough introduction with examples on fiber bundles or a link to a document that deals with it. I was desperately looking on the web for something interesting but I either find 200 pages notes or a poor paragraphe.
Again thank you for being always kind and helpful :)
 A: Are you looking for an introduction to fiber bundles themselves or to methods of computing the cohomology of the various pieces?
The comments already have some very good recommendations on sources for learning about what a fiber bundle is. As far as their topology goes, I suppose I'll mention two facts:


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*The first is about the Euler characteristic and actually follows in some ways from the next point. In fact, one only needs a fibration $p : E \rightarrow B$ here, although some technical conditions are needed: $B$ should be path-connected and the fibration should be orientable over a field. Let $F$ be the fiber of the fibration. Then $\chi(E) = \chi(B)\chi(F)$. 


Here's an example of this in practice. Identify $S^3$ with $SU(2)$ and implement the Hopf fibration $S^1 \rightarrow S^3 \rightarrow S^2$. The subgroup $U(1)$ is realized as $S^1$ and the quotient $SU(2)/U(1)$ is realized as $S^2$. This gives rise to a fibration $SU(2) \rightarrow SU(2)/U(1)$ with fiber $U(1)$; hence $\chi(SU(2)) = \chi(S^2)\chi(S^1) = 0$. 
(Note that this can be seen also by observing that $SU(2)$ is a compact Lie group of positive dimension; hence it admits a nowhere vanishing vector field and must have Euler characteristic zero.)


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*The technical tool in the background here is the homology spectral sequence. Instead of trying to write out all the details here, let me just point you to Hatcher's introduction: http://www.math.cornell.edu/~hatcher/SSAT/SSch1.pdf
