指标定理（二） (10-1)

注意：指标定理（2/5）

6

are G-equivariant, where G acts on U × G on the right by

(x，h) · g＝(x，hg).

For a vector bundle η : E → M，let Fr(E)＝⊔Fr(Eₓ) be the collection of all frames in the fibers of E，and π：Fr(E) → M be the map sending Fr(Eₓ) to x. Fr(Eₓ) can be identified with GL(n，ℝ)，and GL(n，ℝ) acts on Fr(E) on the right. This makes it into a principal GL(n，ℝ)-bundle.

The transition maps of π：Fr(E) → M are the same as those of η： E → M. Thus from a principal GL(n，ℝ)-bundle，we can recover the associated vector bundle using those transition maps.

Let π：P → M be a principal G-bundle and ρ： G → GL(V) a representation of G on a finite-dimensional vector space V. We write ρ(g)υ as g · υ or even gυ.The associated bundle E：＝ P × ᵨ V is the quotient of P × V by the equivalence relation

(p，υ) ~ (pg，g⁻¹ · υ) for g ∈ G and (p，υ) ∈ P × V

We denote the equivalence class of (p，υ) by [p，υ]. The associated bundle comes with a natural projection β：P × ᵨ V → M，β (l，[p，υ])＝π(p).

Theorem 7 If EG → BG is α principαl G-bundle such thαt EG is homotopicαllu triviαl，then for αn αrbitrαry principαl G-bundle π：E → B oυer α simpliciαl complex B (or more generαlly，α CW-complez)，there’s α mαp f：B → BG such thαt f*EG＝E. Equiυαlently，there’s α homomorphism F：E → EG.

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