Adams operation and the boundary pushforward of a sphere bundle

For

p_{!} : K^{- 1} (S (E)) \to K^{0} (X)

in the K-theory Gysin sequence,

p_{!} (ψ^{k} x) = ρ^{k} (E) ψ^{k} (p_{!} x) .

This follows by applying

ψ^{k}

to

δ x = λ_{E} p_{!} (x)

.

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- Second Adams operation on the odd K-theory of the sphere bundle of two tautological lines Adams operation and the boundary pushforward of a sphere bundle

Second Adams operation on the odd K-theory of the sphere bundle of two tautological lines ( $ψ^{2} : K^{- 1} (S (γ \oplus γ)) \to K^{- 1} (S (γ \oplus γ))$ )

Choose

a, b

with

p_{!} (a) = t^{n - 1}

and

p_{!} (b) = t^{n}

, where

t = 1 - [\overline{γ}]

. Then

ψ^{2} (a) = 2^{n + 1} a - (n + 1) 2^{n} b, ψ^{2} (b) = 2^{n + 2} b .

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 142 / 4 / Solution

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