 K-theory Euler class of the tangent bundle of complex projective space

ID: k-theory-euler-class-of-the-tangent-bundle-of-complex-projective-space

Let

γ

be the tautological line bundle and put

t = 1 - [γ]

T CP^{n} \oplus C ≅ (n + 1) \overline{γ} .

Hence

λ_{s} (\overline{T CP^{n}}) = \frac{( 1 + s γ ) ^{n + 1}}{1 + s},

and evaluating at

s = - 1

K^{0} (CP^{n}) = Z [t] / (t^{n + 1})

gives

e^{K} (T CP^{n}) = (n + 1) [γ] (1 - [γ])^{n} = (n + 1) t^{n} .

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