In the following link : http://math.uchicago.edu/~may/REU2017/REUPapers/Malaney.pdf , page : $13$ , there is a paragraph, which says :
When looking at complex characteristic classes with coefficients in $\mathbb{Z}$, there exist analogues of Stiefel-Whitney classes called Chern classes. Once again we will skip the construction, but they can be defined axiomatically as follows :
- Definition $7.5.$ There exist characteristic classes $c_i$ of degree $2i$, for all $i > 0$ which are
characterized by the following axioms :
$ 1) $ $c_0 ( \xi ) = 1 $ for any vector bundle $\xi$.
$ 2) $ If $\xi$ is an $n$ -dimentional vector bundle, then : $c_i (\xi ) = 0$ for $i > n$.
$ 3) $ $ c_k ( \xi \oplus \nu ) = \displaystyle \sum_{ i = 0}^{k} c_i ( \xi ) \smallsmile c_{k-i} ( \nu ) $
$ 4) $ For the canonical line bundle $ \gamma_{ \mathbb{C} }^n $ over $ BU(1) $ , $ c_1 ( \gamma_{ \mathbb{C} }^n ) $ is a fixed generator of $ H^2 ( BU(1) )$. We could also require that $c_1 ( \gamma_{ \mathbb{C} }^n ) $ be the Euler class of the underlying $2$-dimensional real vector bundle. We will define the Euler class later in this section.
- I would like to ask you about the third point of the definition above :
$ 3) $ $ c_k ( \xi \oplus \nu ) = \displaystyle \sum_{ i = 0}^{k} c_i ( \xi ) \smallsmile c_{k-i} ( \nu ) $
Why do we put this condition 3) in this definition ? In other words, why do we want that the Chern classes check this condition 3) ? What is its origin ? From where does it come ?
- Another question that i hope to understand :
How do we find or construct explicitly the Chern classes : $ c_k $ for $ k=0,1, ... , n $ ?
Thanks in advance for your help.