指标定理（一） (11-9)

Let Vectⁿℝ(B) denote the set of isomorphism classes of n-dimensional real vector bun-dles over B，[B，Gₙ] denote the set of homotopy classes of maps B → Gₙ，then we have a surjection

Vectⁿℝ(B) → [B，Gₙ]

Moreover，if B is a CW-complex (or simplicial complex)，then this correspondence is in fact one-to-one.

Theorem 5 Let ξ be α υector bundle oυer B，f₀，f₁：B' → B αre continuous mαps such thαt f₀ is homotopic to f₁.Then if B' is α CW-complex，ωe hαυe f*₀(ξ) ≅ f*₁(ξ).

The proof is done by induction on the skeleta of B'.

Therefore for a CW-complex B we have

₁：₁

Vectⁿℝ(B) ↔ [B，Gₙ(ℝ∞)]，and similar-ly

₁：₁

Vectⁿℂ(B) ↔ [(B，Gₙ(ℂ∞)]. In this sense，Gₙ is called the classifying space for n-dimensional vector bundles，γⁿ is called the universal n-plane bundle，and f：B → Gₙ is called the classifving map for ξ.

This is true for general paracompact space. See Theorem 1.6 in Vector Bundles αnd K-Theory by Hatcher. We sketch the proof in the case of a compact，Hausdorff base.

Theorem 6 Let X be compαct Hαusdorff. Let E

ᴘ

→ B be α υector bundle αnd f₀，f₁： X → B homotopic mαps. Then f*₀(E) ≅ f*₁(E).

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