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.

Equality in K⁰ is stable isomorphism over compact bases

Statement

Assume AC. For a compact Hausdorff space $X$ ,

$[E] - [F] = [E^{'}] - [F^{'}] in K^{0} (X)$

if and only if there is a finite-rank bundle $H$ such that

$E \oplus F^{'} \oplus H ≅ E^{'} \oplus F \oplus H .$

Equivalently, there is an $N \geq 0$ such that

$E \oplus F^{'} \oplus ε^{N} ≅ E^{'} \oplus F \oplus ε^{N} .$

In particular, $[E] = [F]$ if and only if $E \oplus ε^{N} ≅ F \oplus ε^{N}$ for some $N$ .

Facts & Assumptions

Given: AC, a compact Hausdorff space $X$ , and finite-rank complex bundles $E, F, E^{'}, F^{'}$ over $X$ .

[F1]

Equality in the Grothendieck group is the common-summand relation (Complex topological K⁰ by Grothendieck completion).

[F2]

Under AC, every finite-rank bundle over a compact Hausdorff base has a finite-rank complement in a trivial bundle (Finite-rank complement theorem over compact Hausdorff bases).

[A1]

AC is used only through [F2] to obtain the complement.

Proof

technique · direct

1.1

By [F1], the first displayed equality holds exactly when some bundle $H$ satisfies the first stable-isomorphism display. This proves both directions of the first equivalence, including $H = 0_{X}$ when no added summand is needed.

2.1

Apply [F2] to $H$ . There are a bundle $H^{'}$ and $N \geq 0$ with $H \oplus H^{'} ≅ ε^{N}$ . Adding $H^{'}$ to both sides of the isomorphism in step 1.1 gives the trivial-stabilization display. Conversely, that display is the relation in step 1.1 with $H = ε^{N}$ .

F2A1step 1.1algebra

3.1

Set $F = F^{'} = 0_{X}$ in the proved equivalence. Then $[E] = [F]$ exactly when $E \oplus ε^{N} ≅ F \oplus ε^{N}$ for some $N$ , including $N = 0$ .

step 2.1∎

Depends on

Used by

Normalized clutching data for bundles over X×S² Lemma
Hopf-line calculation of K⁰(S²) Theorem
Reduced K-theory exact sequence of a cofibration Theorem

Dependency tree · two levels

8 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, Proposition 2.1 and following construction (standard reference, not scraped)
May, A Concise Course in Algebraic Topology, Chapter 24 §1 (standard reference, not scraped)