指标定理（四） (9-7)

L[M]＝〈L(p(M))，[M]〉＝〈Lₙ(p₁(M)， . . . ，pₙ(M))，[M]〉

where L is the multiplicative sequence associated to

√x 2²ᵏB₂ₖ x x²

──── ＝∑ ────xᵏ＝1＋─ – ─

tanh(√x) k≥0 (2k)！ 3 45

＋ . . . .

For a 4n-dimensional smooth compact oriented manifold，we define its signature σ(M) to be the signature of the symmetric bilinear form defined on H²ⁿ(M⁴ⁿ；ℚ)，

(α，b) ↦〈α∪b，[M]〉.

20

Theorem 12 For α 4n-dimensionαl smooth compαct oriented mαnifold.

L[M]＝σ(M).

Both sides are algebra homomorphisms from Ω*.⨂ to ℚ (where Ω* is the oriented cobordism ring)，so we only need to check the result on the generators ℂP²ᵏ of Ω* ⨂ ℚ.

H²ⁿ(ℂP²ⁿ；ℚ) is generated by a single element and we easily see that its signature is one.

For a complex vector bundle ξ viewed as a real bundle，

1 – p₁＋p₂ – · · · ＝(1 – c₁＋c₂ – . . . )(1＋C，c₁＋c₂＋ . . . ).

In particular，for the tangent bundle of ℂPⁿ，

1 – p₁＋p₂ – · · · ＝(1 – α)ⁿ⁺¹(1＋α)ⁿ⁺¹＝(1 – α²)ⁿ⁺¹，

1＋p₁＋p₂＋ · · · ＝(1＋α²)ⁿ⁺¹. Consequently for ℂP²ᵏ，L(p)＝

α

(──)²ᵏ⁺¹，

tαnh(α)

where α is a generator of the cohomology ring.

To calculate〈L(p(ℂP²ᵏ))，[ℂP²ᵏ]〉，it suffices to calculate the coefficient of α²ᵏ

α

in(───)²ᵏ⁺¹.

tαnh(α)

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