指标定理（二） (10-6)

Of course，there are other ways to introduce characteristic classes，but we are not going to discuss here.

1.6 K-theory

We are assuming the connectedness of the base space so that the dimension of a vector bundle is well-defined. Let V(X) be the set of isomorphism classes of vector bundles over X. This is a semigroup under direct sum. We say that two pairs of vector bundles (E，F)，(E'，F') are equivalent if ∃A ∈ V(X)，E ⨁ F' ⨁ A = E' ⨁ F ⨁ A. The set K(X) of equivalence classes is now a group，called the K-group. V(X) is clearly identified with a subset of K(X).

K(X) is functorial. Given f：X → Y，we have pullback f*：K(Y) → K(X). If (X，*)is a pointed space，f：* → X induces a surjection f*：K(X) → K(*)＝ℤ，whose kernel is defined to be the reduced K-group ˉK(X). Both reduced and unreduced K-groups are rings with multiplication given by the tensor product.

The sequence 0 → ˉK(X) → K(X) → ℤ → 0 splits，so K(X)＝ˉK(X) ⨁ ℤ.

Lemma 5 If bαse spαce X is compαct，then ∀ υector bundle E，∃F such thαt E ⨁ F is triυiαl.

Consider the map α：V(X) → ˉK(X) ⊂ K(X)，E ↦ [E，dimE]， where we use a mumber to denote a trivial bundle. Then α(E)＝α(F) if and only if ∃ trivial bundles k，l such that E ⨁ k＝ F ⨁ l. We call two bundles stαbly equiυαlent if the above condition is satisfied. Also α is surjective since [E，F]＝[E ⨁ A，n] for some A by the lemma. This gives another characterization of ˉK(X).

These statements are true for spaces having the homotopy type of a compact space.

9

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