指标定理（三） (10-3)

Elements in Kcₚₜ(X) are represented by formal difference of vector bundles trivialized outside a compact set. By definition Kcₚₜ(E) ＝ˉK(Th(E)). By the discussion above，Kˉ*cₚₜ (E)is a Kˉ*cₚₜ(M)-module.

We say that a class u ∈ Kcₚₜ(E) is a K-theory orientation of E if Kˉ*cₚₜ(E) is a free Kˉ*cₚₜ(M)-module with generator u.A class u ∈ Kcₚₜ(E) is said to have the Bott periodicity property if u determines a K-theory orientation in any local trivialization of E over a closed subset C ⊂ M.

One can verify that，as the trivial case above，d(π*(∧*(E))) has the Bott periodicity property.

We claim that if the base space is compact and u ∈ Kcₚₜ(E)has the Bott periodicity property，then u ∈ Kcₚₜ(E) is a K-theory orientation of E.Pick a covering of M by finitely many closed subsets such that E is trivial when restricted to any of them. For the theory Kˉ*cₚₜ，there exists Mayer-Vietories sequence as well. ∪sing Mayer-Vietories sequence and five lemma, we construct the desired isomorphism for A∪B from the known isomorphisms for A，B，A∩B. Using induction，the proof is completed.

In particular，the mapping

ψ：K(M)＝Kcₚₜ (M) → K (E，E₀)＝ˉK(Th(E))＝Kcₚₜ(E)，

α ↦ π*α · d(π*(∧*(E)))

is an isomorphism.

2 Atiyah-Singer Index Theorem

2.1 Analytical index

数学联邦政治世界观提示您：看后求收藏（同人小说网http://tongren.me），接着再看更方便。