Chern's Class

Now we follow the same pattern as Stiefel-Whitney class to construct Chern’s class.

Axiom 1.

Axiom 2.

Axiom 3.

Axiom 4.

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Prop. Let $X$ be a compact Hausdorff space. Then

$$\mathrm{Vect}_\mathbb{C}^1(X)\approx[X,\mathbb{C}P^\infty]$$

Proof.

Prop. Let $L_1,L_2$ be two complex line bundle over $X$, then

$$c_1(L_1\otimes L_2)=c_1(L_1)+c_1(L_2).$$

Proof. Let $\pi_i:\mathbb{C}P^\infty\times\mathbb{C}P^\infty\rightarrow\mathbb{C}P^\infty$, $i=1,2$ be two projection maps, and $\boxed{\eta\rightarrow\mathbb{C}P^\infty}$ be the tautological line bundle over $\mathbb{C}P^\infty$. Since $\pi_1^*(\eta)\otimes\pi_2^*(\eta)$ is a line bundle over $\mathbb{C}P^\infty\times \mathbb{C}P^\infty$, then according the the above Proposition, there exists

$$F:\mathbb{C}P^\infty\times \mathbb{C}P^\infty\rightarrow \mathbb{C}P^\infty$$

such that the right square is a pullback diagram.

By Künneth formula,

$$H^2(\mathbb{C}P^\infty\times \mathbb{C}P^\infty)\cong H^2(\mathbb{C}P^\infty)\otimes H^2(\mathbb{C}P^\infty).$$

Let $h\in H^2(\mathbb{C}P^\infty)$ denote the generator. It suffices to prove that

$$F^*h=h\otimes 1+1\otimes h.$$