指标定理（二） (10-8)

We now give another characterization of K(X， Y). Consider the collection 𝓛 (X，Y) of all triples A＝(E，F，σ) where E，F are complex vector bundles over X and σ：E|ʏ → F|ʏ is an isomorphism. Two triples (E，F，σ)， (E'，F'，σ') are said to be isomorphic if there exists bundle isomorphisms ф₁：E|ʏ → E'|ʏ，ф₂：F|ʏ → F'|ʏ such that σ'◦ ф₁＝ф₂◦σ . We say that A＝(E，F，σ)，A'＝(E'，F'，σ') are equivalent if ∃B，B' ∈ 𝓛 (X，Y) such that

A ⨁ B＝A' ⨁ B.

Denote the set of equivalence classes [E，F，σ] by L(X，Y)，and this is an abelian group under the obviously defined direct sum.

Lemma 6 There erists αn equiυαlence of functors χ：L(X，Y) → K(X，Y)， such that ωhen Y＝∅，[E，F，σ] ↦[E] – [F].

We only sketch a proof. Given an element [V₀，V₁，σ] ∈ L₁(X，Y) we associate to it an element χ([V₀，V₁，σ]) ∈ K(X，Y). Set Xₖ＝X × {k} for k＝0，1 and consider the space Z＝X₀ ∪ʏ X₁ obtained from the disjoint union X₀ ∪ X₁ by identifying y × {0} with y × {1} for all y ∈ Y. The natural sequence

ⱼ* ᵢ*

0 → K(Z，X₁) → K(Z) → K(X₁) → 0

10

is split exact since there is an obvious retraction ρ：Z → X₁. Furthermore，there is

≈

an isomorphism φ：K(Z，X₁) → K(X，Y). From our element [V₀，V₁，σ] we define a vector bundle W over Z by setting W[xₖ ≣ Vₖ and identifying over Y via the isomorphism

数学联邦政治世界观提示您：看后求收藏（同人小说网http://tongren.me），接着再看更方便。