指标定理（二） (10-9)

a. Setting W₁ ≡ ρ*(V₁) we have [W]– [W₁] ∈ ker(i*). Hence，there is a unique ele-ment χ([V₀，V₁，σ]) ∈ K(X，Y) with j*φ⁻¹χ([V₀，V₁，σ])＝[W] – [W₁]. This defines the homomorphism χ：L(X，Y) → K(X，Y). ▢

In the discussion above，we are requiring that spaces are compact and CW. But this is not true even for (T*M，T*M₀)，where M is a compact manifold and T*M₀；＝ T*M–zero section. But if we fix a Riemannian metric on T*M and consider bundles D*M，S*M ⊂ T*M with fiber unit solid balls and unit spheres， (D*M，S*M) becomes a CW-pair homotopy equivalent to (T*M，T*M₀). Th(T*M)：＝ D*M/S*M is called the Thom spαce of that bundle.

1.7 Thom isomorphism in K-theory

Recall the Thom isomorphism theorem, stating that for an oriented real rank n bundle π：E → B，there exists a unique class u ∈ Hⁿ (E，E₀；ℤ) such that for all k，we have the Thom isomorphism ф：Hᵏ (B；ℤ) → Hⁿ⁺ᵏ (E，E₀；ℤ)，x ↦(π*x)∪u.

There is a similar version in K-theory.

Theorem 9 For α compler υector bundle π：E → M oυer α compαct spαce M，ωe hαυe αn isomorphism

ψ：K(M) → K(E，E₀)，α ↦ π*α · d(π*(∧*(E))).

Firstly，we explain the element d(π*(∧*(E))) ∈ K(E，E₀). Consider mappings between vector bundles over E，

фᵢ：π*(∧ⁱE) → π*(∧ⁱ⁺¹E)，(ω，υ) ∈ π*(∧ⁱE) ↦ (ω，ω∧υ)

where ω ∈ E. When restricted to E₀， these mappings form an exact sequence，as is easily verified. We claim that these mappings determine a unique element d(π*(∧*(E))) ∈ K(E，E₀).

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