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K-theory Thom class

K-theory Euler class

{c}
{title2=$e^K(T\mathbb{CP}^n)=(n+1)t^n$}

Let $\gamma$ be the tautological line bundle and put $t=1-[\gamma]$. The <Euler sequence on complex projective space> gives
$$
T\mathbb{CP}^n\oplus\mathbb C\cong(n+1)\overline\gamma.
$$
Hence
$$
\lambda_s(\overline{T\mathbb{CP}^n})
=\frac{(1+s\gamma)^{n+1}}{1+s},
$$
and evaluating at $s=-1$ in $K^0(\mathbb{CP}^n)=\mathbb Z[t]/(t^{n+1})$ gives
$$
e^K(T\mathbb{CP}^n)=(n+1)[\gamma](1-[\gamma])^n=(n+1)t^n.
$$


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

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

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K-theory Euler class of the tangent bundle of complex projective space (eK(TCPn)=(n+1)tn)

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

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