PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

K⁰ is contravariantly functorial and homotopy invariant

Statement

Assume AC. A continuous map $f : X \to Y$ of compact Hausdorff spaces induces a unital ring map

$f^{*} : K^{0} (Y) ⟶ K^{0} (X),$

with ${id}^{*} = id$ and $(g \circ f)^{*} = f^{*} g^{*}$ . Homotopic maps induce the same map. If $f$ is based, then $f^{*}$ restricts to ${\tilde{K}}^{0} (Y) \to {\tilde{K}}^{0} (X)$ .

Facts & Assumptions

Given: AC and continuous maps between compact Hausdorff spaces.

[F1]

Pullback bundles have canonical identity and composite comparisons (Vector-bundle pullback is canonically functorial).

[F2]

Under AC, homotopic maps pull a vector bundle back to isomorphic endpoint bundles (Homotopy invariance of vector-bundle pullback).

[F3]

Grothendieck completion is universal for monoid maps (Complex topological K⁰ by Grothendieck completion), and tensor product defines the ring structure (Grothendieck ring structure and rank map).

[F4]

Reduced $K^{0}$ is the kernel of restriction to the basepoint (Reduced complex K-theory).

[A1]

AC is used only through the endpoint-isomorphism theorem [F2].

Proof

technique · direct

1.1

Pullback sends $[E]$ to $[f^{*} E]$ and preserves Whitney sums. By [F3] it extends uniquely to $f^{*} ([E] - [F]) = [f^{*} E] - [f^{*} F]$ . Pullback also preserves tensor products and the trivial line, so this is a unital ring map. The canonical isomorphisms in [F1] give the identity and contravariant composition laws on bundle generators, hence on all virtual classes.

F1F3algebra

2.1

If $f_{0} ≃ f_{1}$ , [F2] gives $f_{0}^{*} E ≅ f_{1}^{*} E$ for every bundle $E$ . The two induced maps therefore agree on all generators and, by the formula in step 1.1, on $K^{0} (Y)$ . This is the sole use of AC.

F2A1step 1.1

3.1

If $f : (X, x_{0}) \to (Y, y_{0})$ is based, then $f \circ i_{x_{0}} = i_{y_{0}}$ . Step 1.1 gives $i_{x_{0}}^{*} f^{*} = i_{y_{0}}^{*}$ , so $f^{*}$ carries the kernel in [F4] into the corresponding kernel.

F1F4step 1.1∎

Depends on

Used by

External product in complex K-theory Definition
Complex K-theory is a two-periodic generalized cohomology theory Theorem
Reduced K-theory exact sequence of a cofibration 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, §2.1 (standard reference, not scraped)
May, A Concise Course in Algebraic Topology, Chapter 24 §1 (standard reference, not scraped)