数学联邦政治世界观
超小超大

指标定理(三) (10-1)

注意:指标定理(3/5)

11

The set of all equivalence classes [V₀,. . .,Vₙ;σ₁,. . .,σₙ] in 𝓛 ₙ(X。Y) will be denoted by Lₙ(X,Y). The set Lₙ(X,Y) is an abelian group under the operation ⨁. Consider the nat-ural map 𝓛 ₙ(X,Y) → 𝓛 ₙ₊₁(X,Y) which associates to each element (V₀,. . .,Vₙ;σ₁,. . .,σₙ) the trivially extended element (V₀,. . .,Vₙ,0;σ₁,. . .,σₙ,0). One verifies that for n ≥ 1 this defines an isomorphism. Consequently everything reduces to the case n=1 which has been discussed above.

Now we have found the unique element d(π*(∧*(E))) ∈ K(E,E₀) determined by фᵢ:π*(∧ⁱE) → π*(∧ⁱ⁺¹E),(ω,υ)∈π* (∧ⁱE) ↦ (ω,ω∧υ).

Secondly,what is the ”product” operation? Indeed K(E,E₀) is a K(M)-module.

We consider more generally. A ring homomorphism μ:K(X) ⨂ K(Y) → K(X × Y)can be defined by μ(α ⨂ b)=p*₁(α)p*₂(b) where p₁ and p₂ are the projections of X × Y onto X and Y. We call it the externel product,and write α * b=μ(α ⨂ b)=p*₁(α)p*₂(b).

We would like to define a similar notion for reduced groups.

Let X∧Y=X × Y/X ∨ Y be the smash product of X,Y. We have the long exact sequence

· · · → ˉK(S(X × Y)) → ˉK(S(X∨Y)) → ˉK(X∧Y) → ˉK(X × Y) → ˉK(X∨Y) → . . .

where the last map in the sequence is a split surjection,with splitting ˉK(X) ⨁ ∼K (Y) → ∼K(X × Y),(α,b) ↦p*₁(α)+p*₂(b) where p₁ and p₂ are the projections of X × Y onto X and Y. Thus the sequence is broken up into short exact sequences. In particular

0 → ˉK(X∧Y) → ˉK(X × Y) → ˉK(X∨Y) → 0

is exact.

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