指标定理（二） (10-10)

Let's generalize the disgussion in the last section. Assume Y ⊂ X is a closed subspace. For each integer n ≥ 1，consider the set 𝓛 ₙ (X，Y) of elements V＝(V₀，V₁，. . .，Vₙ；σ₁，. . .，σₙ) where V₀，. . .，Vₙ are vector bundles on X and where

σ₁ σ₂ σₙ

0→V₀|ʏ → V₁|ʏ → · · · → Vₙ|ʏ → 0

is an exact sequence of bundle maps for the restriction of these bundles to Y. Two such elements V＝(V₀，. . .，Vₙ；σ₁，. . .，σₙ) and V'＝(V'₀，. . .，V'ₙ；σ'₁，. . .，σ'ₙ) are said to be isomorphic if there are bundle isomorphisms φᵢ：Vᵢ → V'ᵢ over X such that everything commutes.

An element V＝(V₀，. . .，Vₙ；σ₁，. . .，σₙ) is said to be elementary if there is an i such that Vᵢ＝Vᵢ₋₁，σᵢ＝id and Vⱼ＝{0} for j ≠ i or i – 1. There is an operation of direct sum ⨁ defined on the set 𝓛 ₙ (X，Y) in the obvious way. Two elements V，V' ∈ 𝓛 ₙ(X，Y) are defined to be equivalent if there exist elementary elements E₁，. . .，Eₖ，F₁，. . .，Fₗ ∈ 𝓛 ₙ (X，Y) and an isomorphism

V ⨁ E₁⨁ · · · ⨁ Eₖ ≅ V' ⨁ F₁ ⨁ · · · ⨁ Fₗ

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