指标定理（一） (11-3)

Let G be a topological group which acts effectively on a space F on the left.A surjection π：E→ B between topological spaces is a fiber bundle with fiber F and structure group G if B has an open cover {∪α} such that there are homeomorphisms

ф：E|∪α → ∪α × F

and the transition functions are continuous functions with values in G：

gαᵝ(x)＝фα фᵦ⁻¹|{x} × ғ ∈G

Sometimes the total space E is referred to as the fiber bundle. A fiber bundle with structure group G is also called a G-bundle. If x ∈ B，the set E᙮=π⁻¹(x) is called the fiber at x. Here the action of a group G on a space F is said to be effective if the only element of G which acts trivially on F is the identity.

A vector bundle of rank n is a fiber bundle with fber ℝⁿ and structure group GL(n，ℝ). If the fiber is ℂⁿ and the structure group is GL(n，ℂ),the vector bundle is a complex vector bundle. Unless otherwise stated,by a vector bundle we mean a C∞ real vector bundle.

Consider two vector bundles ξ and η over the same base space B. ξ is isomorphic to η. written ξ≅η，if there exists a homeomorphism f：E(ξ) → E(η) between the total spaces which maps each vector space Fb(ξ) isomorphically onto the corresponding vector space Fb(η).

Examples include the trivial bundle with total space B × Rⁿ，the tangent bundle τᴍ of a smooth manifold M，and the normal bundle ν of a smooth manifold M ⊂ Rⁿ.An important example，the Grassmann manifold，will be introduced in the next section.

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。