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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