Genera, the Chern Character and the Thom Isomorphism

Proposition 1.2 (Splitting Principle). Let E → M be a complex vector bundle of rank n over a manifold X. Then there is a manifold F (E) and a smooth fibration π : F (E) → M such that • The homomorphism π∗ : H∗(M, Z) → H∗(F (E), Z) is injective. • The bundle π∗E splits into a direct sum of complex line bundles, i.e. π∗E = l1 ⊕ · · ·⊕ ln Proof. Consider the projectivisation p : PC(E) → M and the complex bundle p∗E → PC(E). We have p∗E = PC(E) ×M E and this bundle admits the section l %→ (l, 1l) over PC(E), i.e. there is a line subbundle l1 of p∗(E). Choosing a metric we may decompose p∗(E) = l1 ⊕ l⊥ 1 . Proceeding inductively for l⊥ 1 , we decompose p∗(E) as described. !