数学联邦政治世界观
超小超大

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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