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.

Prop. Let XX be a compact Hausdorff space. Then



Prop. Let L1,L2L_1,L_2 be two complex line bundle over XX, then

c1(L1L2)=c1(L1)+c1(L2).c_1(L_1\otimes L_2)=c_1(L_1)+c_1(L_2).

Proof. Let πi:CP×CPCP\pi_i:\mathbb{C}P^\infty\times\mathbb{C}P^\infty\rightarrow\mathbb{C}P^\infty, i=1,2i=1,2 be two projection maps, and ηCP\boxed{\eta\rightarrow\mathbb{C}P^\infty} be the tautological line bundle over CP\mathbb{C}P^\infty. Since π1(η)π2(η)\pi_1^*(\eta)\otimes\pi_2^*(\eta) is a line bundle over CP×CP\mathbb{C}P^\infty\times \mathbb{C}P^\infty, then according the the above Proposition, there exists

F:CP×CPCPF:\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,

H2(CP×CP)H2(CP)H2(CP).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 hH2(CP)h\in H^2(\mathbb{C}P^\infty) denote the generator. It suffices to prove that

Fh=h1+1h.F^*h=h\otimes 1+1\otimes h.