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

How statement and proof provenance work

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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

Used by

Negative-degree complex K-groups Definition
K-theory of a point and the empty space Example
K⁰ is contravariantly functorial and homotopy invariant Proposition
Reduced K-theory exact sequence of a cofibration Theorem

Dependency tree · two levels

4 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, §2.1 (standard reference, not scraped)
May, A Concise Course in Algebraic Topology, Chapter 24 §1 (standard reference, not scraped)