指标定理（二） (10-3)

We mention some important examples. The universal ℤ₂-bundle is S∞ → ℝP∞. The universal S¹-bundle is S∞ → ℂP∞，induced from S²ⁿ⁻¹ → ℂPⁿ.The Grassmann mani-fold Gₙ (ℝⁿ⁺ᵏ) is a quotient of Vₙ (ℝⁿ⁺ᵏ). Letting k → ∞，we get the universal GL(n，ℝ)-bundle Vₙ(ℝ∞) → Gₙ(ℝ∞). Replacing Vₙ(ℝⁿ⁺ᵏ) by the subset of all orthogonal frames

V⁰ₙ(ℝⁿ⁺ᵏ)＝O(n＋k)/O(k)，we get the universal O(n)-bundle V⁰ₙ(ℝ∞) → Gₙ(ℝ∞).

7

If H ⊂ G is a subgroup. EG → BG is the universal G-bundle，then we have the universal H-bundle EG → EG/H. Since every compact Lie group can be embedded into

O(k) for some k，for compact Lie groups we know the existance of universal bundles.

1.5 Characteristic classes

We may identify a real vector bundle with its associated principal GLₙ(ℝ)-bundle, or Oₙ-bundle if a metric is provided，or SOₙ-bundle if it’s additionally orientable，and similarly for complex case.

First we define the Chern class. We have that

H*(ℂP∞；ℤ)＝H*(BT¹：ℤ)＝ℤ[t].

Thus

H*(BTⁿ；ℤ)＝ℤ[t₁，. . .，tₙ].

For the inclusion ρ： BTⁿ → B∪ₙ，we have a homomorphism

ρ*：H*(B∪ₙ：ℤ) → H*(BTⁿ；ℤ).

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。