指标定理（四） (9-6)

with coefficients in ∧ such that，if the variable xᵢ is assigned degree i，then each Kₙ (x₁，. . .，xₙ)is homogeneous of degree n. Given an element α ∈ AΠ with leading term 1， define a new element K(α) ∈ AΠ by the formula

K(α)＝1＋K₁(α₁)＋K₂(α₁，α₂)＋ . . .

Definition 6 These Kₙ form α multiplicαtiυe sequence if K(αb)＝K(α)K(b) for αll grαded ∧-αlgebrαs A* αnd αll α，b ∈ AΠ₀(i.e.K：AΠ₀ → AΠ₀ is α group homomorphism).

Theorem 11 Giυen α formαl pοωer series f(t)＝1＋λ₁t＋ . . . ωith coefficients in ∧，there ezists α unigue multiplicαtiυe sequence such thαt K(1＋t)＝f(t) for αll 1＋t ∈ AΠ₀.

We omit the proof. Note that if α＝(1＋t₁) . . . (1＋tₙ)，then K(α)＝f(t₁) . . . f(tₙ). tᵢ

Consequently，the Todd class ∏ᵢ ───

1–e⁻ᵗⁱ

∈ HΠ(B∪ₙ；ℚ) is indeed given by the mul-tiplicative sequence associated to the series

x

td(x)＝───，

1–e⁻ˣ

where we take ∧＝ℚ，AΠ＝HΠ(B∪ₙ；ℚ). We may write

Td(c)＝∏ td(tᵢ) ∈ HΠ(B∪ₙ；ℚ).

where c is the total Chern class and tᵢ’ s are the Chern roots.

Here you may notice somet hing strange.tᵢ is indeed in H²(B∪ₙ；ℚ) but not H¹(B∪ₙ；ℚ). To avoid this，we may replace H*(B∪ₙ；ℚ) by H²*(B∪ₙ；ℚ)：＝ ⨁ₖ H²ᵏ(B∪ₙ；ℚ) with grad-ing given by k. We will take this for granted when considering Chern classes，and use H⁴* when considering Pontrjagin classes.

4.2 Hirzebruch signature theorem

We define the L-genus of a 4n-dimensional smooth compact oriented manifold to be

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