DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Reduced complex K-theory

Definition

Let $(X, x_{0})$ be a based compact Hausdorff space, and let $i_{x_{0}} : * \to X$ select the basepoint. Restriction to the fiber gives a unital ring map

$i_{x_{0}}^{*} : K^{0} (X) ⟶ K^{0} (*) ≅ Z,$

where complex dimension identifies $K^{0} (*)$ with $Z$ . The reduced complex K-group is

${\tilde{K}}^{0} (X) = \ker i_{x_{0}}^{*} .$

Thus a virtual bundle belongs to ${\tilde{K}}^{0} (X)$ exactly when its virtual rank is zero on the path component containing $x_{0}$ . Its rank on another component can be different. Since $i_{x_{0}}^{*}$ is a ring homomorphism by Grothendieck ring structure and rank map, ${\tilde{K}}^{0} (X)$ is an ideal in $K^{0} (X)$ .

The empty space has no basepoint, so this based reduced group is not invoked for $X = \emptyset$ .

Depends on

Grothendieck ring structure and rank map

Reduced complex K-theory

Definition

Depends on

Dependency tree · 2 levels

Sources