calculating Euler classes

Question

I want to understand how to compute Euler classes, what are the canonical examples of vector bundles from which i can start, and are there any books or lectures which describe how to compute Euler classes.

Answer 1

If you are familiar with differential forms and de Rham cohomology, then, I would suggest, Bott and Tu's classic Differential Forms in Algebraic Topology. It has a chapter on characteristic classes.

The main tool in computation of characteristic classes is the splitting principle, which allows us to pretend that any bundle is direct sum of line bundles (for the purpose of computation of characteristic classes). Bott&Tu have worked out many examples of computing Euler class of operations on vector bundles like dual, direct sum, tensor product, symmetric power, exterior power etc.

Another fact to keep in mind is that Euler class is Poincaré dual of the zero set of a generic section, where by, 'generic' I mean a section that is transverse to the zero section. For example, the Euler class of the canonical line bundle over $\mathbb{CP}^n$ is the Poincaré dual of the hyperplane class $[H] \in H_{2n-2}(\mathbb{CP}^n)$.

Lastly, another useful fact is that for complex bundles, the top Chern class is the Euler class (so you can use the axioms and results for Chern class to compute the Euler class). And for real bundle the natural map $H^{\text{top}}(B, \mathbb{Z}) \to H^{\text{top}}(B, \mathbb{Z}_2)$ maps the Euler class to the top Stiefel-Whitney class.

If you are more comfortable with singular cohomology, then I would suggest Milnor and Stasheff's Characteristic Classes. They have worked out the properties and examples in the singular cohomology setting. (This book was recently LaTeXified by a group of students, you can find it here.)

One standard example is that of the tangent bundle of $\mathbb{CP}^n$ and is worked out in both the books.