K-theory Euler class ( $e^{K} (E) = Λ_{- 1} (\overline{E})$ )

Pulling the K-theory Thom class back along the zero section gives

e^{K} (E) = Λ_{- 1} (\overline{E}) = \sum_{j} (- 1)^{j} [Λ^{j} \overline{E}] .

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

K-theory Euler class of the tangent bundle of complex projective space ( $e^{K} (T CP^{n}) = (n + 1) t^{n}$ )

Let

γ

be the tautological line bundle and put

t = 1 - [γ]

. The Euler sequence on complex projective space gives

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

in

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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