指标定理（一） (11-8)

Let ℝ∞ denote the vector space consisting of those infinite sequences x＝(x₁， x₂，x₃， . . .)with only finitely many nonzero coordinates. For fixed k，the subspace consisting of all x＝(x₁，x₂，. . .，Xₖ，0，0，. . .) will be identified with the coordinate space ℝᵏ. The infinite Grassmann manifold Gₙ＝Gₙ(ℝ∞) is the set of all n-dimensional linear sub-spaces of ℝ∞，topologized as the direct limit of the sequence Gₙ(ℝⁿ) ⊂ Gₙ(ℝⁿ⁺¹) ⊂ Gₙ(ℝⁿ⁺²) ⊂. . . . As a special case， the infinite projective space P∞＝G₁(ℝ∞) is equal to the direct limit of the sequence P¹ ⊂ P² ⊂ P³ ⊂. . . .

A canonical bundle γⁿ over Gₙ is constructed， just as in the finite dimensional case，as follows. Let E(γⁿ) ⊂ Gₙ × ℝ∞ be the set of all pairs ( n-plane in ℝ∞，vector in that n-plane )，topologized as a subset of the Cartesian product. Definen π：E(γⁿ) → Gₙ by π(X，x)＝X. In fact γⁿ is locally trivial，and we omit the proof.

Recall that a paracompact space is a Hausdorff space such that every open covering has a locally finite open refinement. The infinite Grassmann space is paracompact.

Lemma 4 For αny fiber bundle ξ oυer α pαrαcompαct spαce B，there erists α locαlly finite coυering of B by countably mαny open sets ∪₁，∪₂，∪₃，. . .，so thαt ξ|∪ᵢ is triυiαl for eαch i.

Theorem 3 Any ℝⁿ-bundle ξ oυer α pαrαcompαct bαse spαce αdmits α bundle mαp ξ → γⁿ.

Theorem 4 Any tωο bundle mαps ξ → γⁿ αre bundle-homotopic.

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