指标定理（二） (10-7)

Now we talk about complex bundles. The inclusion Uₙ ⊂ Uₙ₊₁ induces jₙ： BUₙ → BUₙ₊₁.If f is the classifying map for a rank n bundle E，jₙ◦f is the classifying map for E ⨁ 1. E and E ⨁ 1 are stably equivalent. Also jₙ induces (jₙ)*：[X，B∪ₙ] → [X，B∪ₙ₊₁]，and we write [X，B∪] for the direct limit of the direct system consisting of {[X，B∪ₙ]}. Thus we have

[X，B∪]＝ˉK(X).

Theorem 8 ˉK(Sⁿ) is isomorphic to ℤ if n is eυen，αnd 0 ifn is odd.

A generator of ⁻K(S²) is given by H – 1 where H is the hyperplane bundle. The first Chern class sends H to 1∈ ℤ，and hence defines an isomorphism ˉK(S²)＝ℤ.

Define K(X，A)＝ˉK(X/A).where the pair (X，A) is assumed to satisfy the homotopy extension property，e.g. a CW-pair. Then we have exact sequences

K(X，A) → ˉK(X) → ˉK(A)，

K(X，A) → K(X) → K(A).

We use SX，CX to denote the reduced suspension of X and the reduced cone over X respectively. Then we have exact sequences

ˉK(SA) → ˉK(X∪CA) → K(X)，

· · · → ˉK(S(X/A)) → ˉK(SX) → ˉK(SA) → ˉK(X/A) → ˉK(X) → ˉK(A).

Define ˉK⁻ⁿ(X)＝ˉK(SⁿX). Periodicity theorem tells us that ˉK(S²X)＝ˉK(X) holds for arbitrary space X，and hence we can also extend the ”Puppe sequence” to the right.

Also there is an unreduced version

· · · → Kⁿ(X，A) → Kⁿ (X) → Kⁿ (A) → Kⁿ⁺¹ (X，A) → Kⁿ⁺¹(X) → Kⁿ⁺¹ (A) → . . . .

K-theory is a generalized cohomology theory.

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