指标定理（二） (10-2)

Proof. We may assume that E is trivial when restricted to each simplex. We use induction. The result is clear on 0-skeleton. Suppose now we have F|ᴇᵏ：Eᵏ＝π⁻¹(Bᵏ) → EG. For a (k ＋1)-simplex σ of B，F|π⁻¹(∂σ) is determined by F|∂σ×e：∂σ × e → E. Since EG is homotopically trivial，F|∂σ×e can be extended to F|σ×e，and then to F|π⁻¹(σ) by G-equivariance. ▢

Such a bundle is called the universal G-bundle，and BG is called the classifying space of G.

Also，the map defined above is determined up to homotopy. Given two homomor-phisms F₀，F₁：E → EG，consider the bundle l × E → I × G. F₀，F₁ gives a map ∂l × E → EG，so we can extend the map to l × E → EG as above.

Write P(B，G) for the collection of (isomorphism classes of) principal G-bundles over B. By the discussion above we have a surjective mapping

P(B，G) → [B，BG].

Indeed it’s bijective，and this is why BG is called the classifying space. Bijectivity is proved using covering homotopy property.

If there is another universal G-bundle E'G → B'G，there are induced maps f：BG → B'G，g：B'G → BG such that fg，gf are homotopic to identity. Thus the classifying space，if exists，is determined up to homotopy equivalence. In particular，H*(BG；R) is completely determined.

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。