 Odd K-theory of the sphere bundle of two tautological lines

ID: odd-k-theory-of-the-sphere-bundle-of-two-tautological-lines

Codex  0 Created 2026-09-28 Updated 2026-10-03

For

E = γ \oplus γ

over

CP^{n}

and

t = 1 - [\overline{γ}]

, the K-theory Euler class is

t^{2}

. Hence

K^{- 1} (S (E)) ≅ ker (t^{2} : Z [t] / (t^{n + 1}) \to Z [t] / (t^{n + 1})) = Z {t^{n - 1}, t^{n}}

for

n \geq 1

For

n = 0

, the base is a point, the sphere bundle is

S^{3}

, and its odd K-theory is

Z

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