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Complex K-theory of spheres

Statement

Assume AC. For $m \geq 0$ ,

${\tilde{K}}^{0} (S^{2 m}) ≅ Z, {\tilde{K}}^{0} (S^{2 m + 1}) = 0.$

The even generator is the $m$ -fold reduced external product of the Bott class $β$ , with $S^{0}$ interpreted as a based two-point space and the empty product as its rank-difference generator. Equivalently, ${\tilde{K}}^{q} (S^{n})$ is $Z$ when $q - n$ is even and is zero when $q - n$ is odd.

Facts & Assumptions

Given: AC, based spheres, and the Hopf Bott class $β$ .

[F1]

Multiplication by $β$ is the natural twofold-suspension isomorphism in every degree (Complex Bott periodicity).

[F2]

Complex bundles on $S^{1}$ are classified by clutching data on $S^{0}$ and ${GL}_{n} (C)$ is path-connected in the complex case (Clutching classifies vector bundles over spheres in the stable range).

[F3]

${\tilde{K}}^{0} (S^{2}) = Z β$ (Hopf-line calculation of K⁰(S²)).

[A1]

AC is propagated from [F1] and [F3]; the $S^{0}$ and $S^{1}$ base calculations themselves are finite and choice-free.

Proof

technique · direct

1.1

A bundle on the based two-point space $S^{0}$ is a pair of finite-dimensional complex vector spaces. The reduced kernel records the dimension at the nonbasepoint minus the dimension at the basepoint, so ${\tilde{K}}^{0} (S^{0}) ≅ Z$ . On $S^{1}$ , [F2] reduces a rank- $n$ bundle to two clutching values in the same path component of ${GL}_{n} (C)$ , so it is trivial. Thus $K^{0} (S^{1}) ≅ Z$ by rank and ${\tilde{K}}^{0} (S^{1}) = 0$ .

F2algebra

2.1

Apply [F1] repeatedly to the two base groups in step 1.1. It gives ${\tilde{K}}^{0} (S^{2 m}) ≅ {\tilde{K}}^{0} (S^{0}) = Z$ and ${\tilde{K}}^{0} (S^{2 m + 1}) ≅ {\tilde{K}}^{0} (S^{1}) = 0$ for all $m \geq 0$ . At each even step the isomorphism is external product with $β$ , so the generator is the stated $m$ -fold product; for $m = 1$ it agrees with [F3].

F1F3A1step 1.1induction

3.1

By definition, suspension shifts the reduced degree and [F1] makes it two-periodic. Hence ${\tilde{K}}^{q} (S^{n})$ depends only on the parity of $q - n$ ; step 2.1 gives $Z$ in even parity and zero in odd parity. This includes $n = 0$ , $m = 0$ , and the zero group without a hidden exception.

F1step 2.1algebra∎

Depends on

Used by

Complex K-theory of even and odd spheres Example
K-theory of a point and the empty space Example
Complex K-theory is a two-periodic generalized cohomology theory Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hatcher, Vector Bundles & K-Theory, Corollary 2.12 (standard reference, not scraped)
May, A Concise Course in Algebraic Topology, Chapter 24 §2 (standard reference, not scraped)