There are two styles for the definition of the Chern classes $c_k(E)$, for a vector bundle $E\rightarrow X$, defined by the axioms:
- $c_0(E)=1$,
- $c_k(f^*E)=f^*c_k(E)$ for a continuous map $f:X\rightarrow Y$,
- $c_k(E\oplus F)=\sum_{i=0}^k c_i(E)\smallsmile c_{k-i}(F)$, and
- $-c_1(L)$ is the generator of $H^2(P_1\mathbb{C})=\mathbb{Z}$, where the canonical line bundle $L=\{(\xi,z)\in P_1\mathbb{C}\times \mathbb{C}^2:z\in \xi\}$ of the Riemannian sphere $P_1\mathbb{C}$,
and defined by the following construction (Grothendieck):
First, we define the class $c_1(L)\in H^2(X)$ for any line bundle $L$ on $X$ (for example, by using Čech cohomology). While, for all vector bundle $E$ on $X$, let $\mathbb{P}(E)$ denote the projective fiber bundle of $E$ with fiber $i:\mathbb{P}(E_x)\hookrightarrow\mathbb{P}(E)$, that is, $\mathbb{P}(E)=E^*/\mathbb{C}^*$ where $E^*=E-\{0\in E_x:x\in X\}$, let $p:\mathbb{P}(E)\rightarrow E$ be the projection and let $$L(E)=\{(l,e)\in p^*E:e\in l\}.$$ Then for $a=-c_1(L(E))\in H^2(\mathbb{P}(E))$, the $1, i^*a, \cdots, i^*a^{n-1}$ form a free basis of $H^*(\mathbb{P}(E_x))$. Therefore we define the classes $c_k(E)\in H^{2k}(X)$ by expanding Leray–Hirsch theorem the class $-a^n$, with the relation: $$a^n+p^*c_1(E)\smallsmile a^{n-1}+\cdots+p^*c_{n-1}(E)\smallsmile a+p^*c_n(E)=0. \cdots(*) $$
My question is: do they follow each other? How do I show it?
I already proved the implication "Grothendieck $\Rightarrow$ the four axioms". But I can't do the contrary. To do it, we should only check the final equation $(*)$. Could you give me any help?