指标定理（二） (10-5)

(–1)ⁱc₂ᵢ(ξ ⨂ ℂ)∈ H⁴ⁱ(B；ℤ). Indeed，H*(BOₙ；ℚ) is the polynomial ring generated by the universal Pontrjagin classes p₁，. . .，pₘ where m＝[n/2]. We sketch a proof. Let T ⊂ Oₙ be the maximal torus in Oₙ. ℝⁿ decomposes into 2-dimensional subspaces Uⱼ，j＝1，. . .，m (and possibly one additional 1-dimensional subspace). After complexifying，each Uⱼ ⨂ ℂ decomposes into Vⱼ ⨁ ˉVⱼ . The total Chern class of the complexified bundle is ∏ⱼ(1–x²ⱼ)，so the total Pontrjagin class is p＝∏ⱼ(1＋x²ⱼ)，i.e.

ρ*：H*(BOₙ：ℚ) → H*(BT；ℚ)，p↦∏(1+x²ⱼ).

ⱼ

8

It follows that H*(BOₙ：ℚ) is the subalgebra of symmetric polynomials of x²₁，. . .，x²ₘ， which gives the conclusion.

Similarly we can calculate H*(BSOₙ；ℚ)， since the maximal torus in Oₙ is indeed in SOₙ. If n＝2m＋1 is odd，H*(BSOₙ；ℚ) is still the subalgebra of symmetric poly-nomials of x²₁，. . .，x²ₘ，i.e. the polynomial ring generated by the universal Pontrjagin classes p₁，. . .，Pₘ. But if n＝2m is even，H*(BSOₙ：ℚ) is the subalgebra of symmetric polynomials generated by x²₁，. . .，x²ₘ₋₁，x₁x₂. . .xₘ. Indeed

H*(BSO₂ₘ：ℚ)＝ℚ[p₁，. . .pₘ₋₁，χ]

where χ is the Euler class defined for oriented real bundles.

Let HΠ(B；ℤ) denote the collection of formal power series. Consider the expression

ₙ

∑ eᵗⁱ ∈ HΠ(BTⁿ；ℚ).

ᵢ₌₁

It’s symmetric so it belongs to HΠ(B∪ₙ；ℚ). We define its pullback in HΠ(B；ℚ) to be the Chern character of π：E → B，denoted by ch(E). Also the pullback of

tᵢ

∏ ─── ∈ HΠ(BUₙ；ℚ) ⊂ HΠ(BTⁿ；ℚ)

ᵢ 1 – e⁻ᵗⁱ

is defined to be the Todd clαss.

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