# 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 :)

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I'd start with trying to understand the tangent bundle and the Hopf fibration thoroughly. These are two good examples that show many of the inherent intricacies. Then I think good place to look at is the book by Bott and Tu on differential forms in algebraic topology. One of the standard references on fibre bundles is Husemöller's book. –  t.b. May 18 '11 at 12:11
The section on fibre bundles in Hatcher's Algebraic Topology is a very good introduction (p375-384), and his Vector Bundles book is fab too (the first chapter, which goes up to classification, is about 35 pages long, but it's a very easy read). You can find both books available free online on Hatcher's website. For more depth, Husemöller (mentioned above by Theo) is excellent. –  Alex May 18 '11 at 13:04
Many thanks Alex and Theo –  El Moro May 18 '11 at 15:02
Steenrod's Topology of Fiber Bundles is a classic that will get your hands dirty. –  Manuel Nov 23 '11 at 2:14

• 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.)