指标定理（二） (10-4)

This is indeed an injection.We can show that its image is invariant under Weyl group，and thus is contained in the subalgebra of symmetric polynomials，which is generated by elementary symmetric polynomials σ₁，. . .，σₙ. Calculating the Poincaré series， we find out that on each degree，H*(B∪ₙ；ℤ) and ℤ[σ₁，. . .，σₙ has the same rank as abelian groups. However，replacing ℤ with arbitrary coefficient field R，we still have an injection ρ*：H*(B∪ₙ：R) → H*(BTⁿ；R)， which implies that

H*(B∪ₙ；ℤ)＝ℤ[σ₁，. . .，σₙ] ⊂ H*(BTⁿ；ℤ)＝ℤ[t₁，. . .，tₙ]，

where σ₁，. . .，σₙ are the elementary symmetric polynomials of t₁，. . .，tₙ.

We define σ₁，. . .，σₙ ∈ H*(B∪ₙ；ℤ) to be the Chern classes of the universal rank n complex vector bundle.

For general complex vector bundle π：E → B (with metric)，by the discussion above，there exists a map f：B → B∪ₙ which pulls back the universal bundle to get E and is determined up to homotopy. For each k，f*σₖ ∈ H*(B；ℤ) is well-defined and we define it to be the k-th Chern clαss of E，denoted by cₖ(E).c(E)：＝1＋c₁(E)＋c₂(E)＋ · · · ∈H*(B；ℤ) is the total Chern class.

Similarly，for the real case we have an injection of algebras

ρ*：H*(BOₙ；ℤ₂)＝ℤ₂[ω₁，. . .，ωₙ] → H* (Bℤⁿ₂；ℤ₂)＝ℤ₂[t₁，. . .，tₙ]

where ω₁，. . .，ωₙ are elementary symmetric polynomials. These ω₁，. . .，ωₙ are called Stiefel-Whitney clαsses.

For a real vector bundle ξ of rank n，we define the i-th Pontrjagin class to be pᵢ(ξ)＝

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