指标定理（一） (11-10)

Proof.Let h：X × l → B be a homotopy from f₀ to f₁.Then f*₀(E)＝h*(E)|x×{0} and similarly for f*₁(E).So without loss of generality we may replace B by X × l，and we wish to show that the time 0 and time 1 restrictions of a bundle E on X × l are isomorphic. Using compactness，one can show that there is a finite cover {U₁，. . .，Uₙ} of X so that the restriction of E to each Uᵢ × l is trivial.Let {φᵢ}ⁿ ᵢ₌₁ be a partition of unity subordinate to the cover {∪ᵢ}ⁿ ᵢ₋₁ . For each 0 ≤ j ≤ n，define Φⱼ＝∑ʲᵢ₌₁ φᵢ. Thus Φ₀＝0 and Φₙ = 1 on X. For simplicity，we will assume n＝2，since that is enough to see the argument. Thus we have

Φ₀＝0 ≤ Φ₁＝φ₁ ≤ Φ₂＝1

on X.For each 0 ≤ j ≤ n，we define Xⱼ ⊆ X × l to be the graph of Φⱼ.Thus X₀＝X × {0} and X₂＝X × {1}，and each Xⱼ is homeomorphic to X νia the projection. Finally，let Eⱼ be the restriction of E to Xⱼ ≅ X. We claim that E₀ ≅ E₁ ≅ E₂. To see that E₀ ≅ E₁，recall that E is trivial on ∪₁ × l. It follows that the trivialization of E on ∪₁ restricts to trivializations φ∪₁ of E₀ and E₁ on ∪₁. Define α∪₁： (E₀)|∪₁ → (E₁) |∪₁ to be the composition

(φ∪₁)|ᴇ₀ (φ∪₁)⁻¹

(E₀)|∪₁ → F × ∪₁ → (E₁)|∪₁

Now let V₁＝X–supp(φ₁). Since φ₁ is supported inside ∪₁，it follows that ∪₁∪V₁＝X. Also，we have that (E₀)|ᵥ₁＝(E₁)|ᵥ₁，since X₀ ∩(V₁ × l)＝V₁ × {0}＝X₁ ∩(V₁ × l). Now α∪₁ on (E₀)|∪₁ glues together with id on (E₀) |ᵥ₁ to give an isomorphism E₀ ≅ E₁. ▢

1.4 Principal bundles and classifying space

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