指标定理（三） (10-2)

If α ∈ ˉK(X)，b ∈ ˉK(Y)，we claim that α * b ∈ K(X × Y) indeed comes from ˉK(X∧Y). This is true since the image of α * b in ˉK(X∨Y) is 0. We define that element in ˉK(X∧Y) to be their externel product，and still denote it by α * b. This reduced externel product

ˉK(X) ⨂ ˉK(Y) → ˉK(X∧Y)

is still a ring homomorphism.

Replace the X above by X⁺ and let Y＝X/A. Here X⁺ is the pointed space(X ⊔ {*}，*). The diagonal map of X induces Δ：X/A → X⁺∧(X/A)，and consequently

Δ*

ˉK(X⁺) ⨂ ˉK(X/A) → ˉK(X⁺∧(X/A)) → ˉK(X/A).

Also ˉK(X⁺)＝K(X). The formula above makes K(X，A) into a K(X)-module，In particular， using π*，K(E，E₀) is a K(M)-module.

By the way，there's a general version of periodicity theorem. β：ˉK (X) → ˉK (S²X)＝ˉK(S²∧X)，α ↦(H – 1) * α， is an isomorphism for all compact Hausdorff spaces X.

Lastly，we talk about its proof.

If M is a point and E is the trivial line bundle， d(π*(∧*(E))) ∈ K(S²) is determined by the sequence on E₀，

0 → E × ℂ → E × E → 0，

where the mapping in the middle sends (ω，υ) to (ω，ω∧υ)＝(ω，υ). It suffices to show that this is a generator of K(S²)＝ℤ. Indeed this is just the generator 1 – H.

The proof uses K-theory with compact support. If X is a locally compact space，and X∞ is its one point compactification，define Kcₚₜ(X)＝ˉK(X∞)，K⁻ⁱcₚₜ(X)＝Kcₚₜ(X × ℝⁱ).

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