TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Complex Bott periodicity

Statement

Assume AC. Reduced external product with $β \in {\tilde{K}}^{0} (S^{2}) = K^{- 2} (*)$ is a natural isomorphism

${\tilde{K}}^{- n} (X) \overset{≅}{\to} {\tilde{K}}^{- n - 2} (X)$

for every $n \geq 0$ and based finite CW complex $X$ , and likewise for compact pairs in the stated category. It extends the grading uniquely to natural isomorphisms $K^{q} (X) ≅ K^{q - 2} (X)$ for all $q \in Z$ .

Facts & Assumptions

Given: AC, a based finite CW complex $X$ , and $β = [γ] - 1 \in {\tilde{K}}^{0} (S^{2})$ .

[F1]

The product theorem gives the unique decomposition $K^{0} (X \times S^{2}) = K^{0} (X) \oplus K^{0} (X) β$ (Fundamental product theorem for complex K-theory).

[F2]

Reduced external product is the unique class on the smash product whose pullback is the unreduced product (External product in complex K-theory).

[F3]

Negative groups are reduced groups of iterated suspensions (Negative-degree complex K-groups).

[F4]

The reduced cofibration sequence is natural and exact at every suspended stage (Reduced K-theory exact sequence of a cofibration).

[A1]

AC is used through the product theorem and the reduced external-product and exactness suppliers [F1], [F2], and [F4].

Proof

technique · direct

1.1F1F4A1algebra

The wedge inclusion $X \lor S^{2} \to X \times S^{2}$ has restriction map split by the two projections. Hence reduced exactness [F4] identifies ${\tilde{K}}^{0} (X \land S^{2})$ with the subgroup of classes on the product restricting to zero on both axes. In the normal form $a + b β$ from [F1], restriction to $X \times {*}$ is $a$ , while restriction to ${x_{0}} \times S^{2}$ is $a (x_{0}) + b (x_{0}) β$ . Both vanish exactly when $a = 0$ and $b \in {\tilde{K}}^{0} (X)$ .

2.1F2F3A1step 1.1

By [F2], the class corresponding to $b$ in step 1.1 is exactly the reduced external product $b ⊠ β$ . Therefore $b \mapsto b ⊠ β$ is a natural isomorphism ${\tilde{K}}^{0} (X) \to {\tilde{K}}^{0} (X \land S^{2}) = {\tilde{K}}^{0} (Σ^{2} X)$ . Taking $X = S^{0}$ with one nonbasepoint sends its rank-difference generator to $β$ on $S^{0} \land S^{2} ≅ S^{2}$ . Since the unbased point has $*_{+} = S^{0}$ , [F3] identifies this target with the coefficient group $K^{- 2} (*)$ .

3.1F2F3step 2.1

Apply step 2.1 to $Σ^{n} X$ . A suspension of a finite CW complex is again a compact based finite CW complex, and $S^{2} \land Σ^{n} X ≅ Σ^{n + 2} X$ . Using [F3] gives the displayed isomorphism ${\tilde{K}}^{- n} (X) ≅ {\tilde{K}}^{- n - 2} (X)$ for every $n \geq 0$ . Replacing $X$ by the compact quotient $X / A$ gives the relative statement; naturality follows from naturality of external product and quotient maps.

4.1step 3.1algebra

Define positive degrees by transporting the already defined nonpositive groups along the inverse of step 3.1: choose $r$ with $q - 2 r \leq 0$ and set $K^{q} (X) = K^{q - 2 r} (X)$ using the canonical composite of inverse Bott maps. If a larger $r$ is used, the two composites differ by a Bott isomorphism followed by its inverse, so the identification is independent of $r$ . This is the unique extension for which multiplication by $β$ gives $K^{q} ≅ K^{q - 2}$ in every degree.

5.1F2F4A1step 3.1step 4.1∎

The maps in [F4] commute with external product by naturality, so the two-periodic identifications respect absolute, reduced, and relative maps and their exact sequences. This proves the stated natural periodic theory, including the zero group and the one-point boundary cases.

Complex Bott periodicity

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources