ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck pending

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.

Integer components of the based S3 gauge group

Example

Take the trivial principal $S U (2)$ bundle over the oriented sphere $S^{3}$ , fix a point and a frame over it, and identify $S U (2)$ with the unit quaternions $S^{3}$ . Use $G_{\infty}$ and ${\overline{G}}_{\infty}$ from the configuration definition, with based smooth subgroup $G_{\infty, m}$ ; for brevity put $G_{m} : = G_{\infty, m}$ . For every integer $k$ , a based map $u_{k} : S^{3} \to S^{3}$ of degree $k$ defines a based smooth gauge transformation after relative smoothing. These transformations represent distinct components of the based gauge group, so $π_{0} (G_{m}) ≅ π_{3} (S^{3}) ≅ Z$ . The full smooth gauge group has the same component set, and quotienting by the two-element center can merge at most two components; both the full and reduced smooth groups therefore have infinitely many path components. This example is specific to the trivial $S U (2)$ bundle over $S^{3}$ and does not establish the $S^{4}$ statement or an arbitrary-rank claim.

Example references

Hatcher, Algebraic Topology, Chapter 4

May, A Concise Course in Algebraic Topology

Facts & Assumptions

Given: The trivial bundle over the oriented $S^{3}$ , the chosen evaluation frame, and the based maps $u_{k}$ in the Example. The actual configuration carrier exists, but its current hash-bound Step 1 record is ready according to loadStep1Record and step1Decision; this is authoring readiness, not independent approval.

[F1]

The based gauge-group components in the trivial or nontrivial bundle sector are $π_{0} (G_{m}) ≅ π_{3} (S U (2))$ for the $S^{3}$ case. Based gauge-group homotopy via hemisphere clutching

[F2]

Unit quaternions identify $S U (2)$ with $S^{3}$ , and degree classifies based self-maps of $S^{3}$ . Low special-unitary homotopy inputs, Based sphere maps are classified by degree

[F3]

The full and reduced gauge groups have infinitely many components in the $r = 3$ case; the finite quotient acts on components with orbits of size at most $2$ . Nontrivial homotopy of reduced sphere gauge groups

[F4]

The actual configuration definition fixes the smooth gauge group $G_{\infty}$ with finite-atlas $C^{\infty}$ section topology, its based subgroup, and the quotient ${\overline{G}}_{\infty} = G_{\infty} / Z_{N}$ , along with the bundle, frame, and right-action conventions. Its carrier was inspected in full, but its current hash-bound Step 1 record is ready according to loadStep1Record and step1Decision; this is authoring readiness, not independent approval. Smooth and Sobolev configuration and reduced gauge spaces

[F5]

Smooth and continuous based gauge sections have the same sphere/disk parameter homotopy classes. Parametric smoothing of gauge sections

[F6]

Paths lift to a covering once their initial lift is fixed. Existence and uniqueness of path lifts through a covering map

[A1]

Degree is normalized by the standard orientations on source and target $S^{3}$ ; the component comparison for the center quotient uses the covering-space topology fixed in [F3].

Verification

1.1A1F1F2F5construct

By [F2], each integer $k$ is represented by a based map $u_{k} : S^{3} \to S U (2)$ of degree $k$ . The standard degree representative may be chosen smooth by a relative smooth approximation; [F5] then identifies its class with one in the smooth based section topology. Distinct integers give distinct elements of $π_{3} (S U (2))$ by degree classification, so [F1] proves that the based gauge group has components indexed by $Z$ .

2.1F1F2F3F4F6algebra∎

The evaluation fibration has connected base $S U (2)$ and $π_{1} (S U (2)) = 0$ by [F2, F3]. Its component LES therefore identifies $π_{0} (G_{m})$ with $π_{0} (G)$ , so the full group also has infinitely many components. The quotient by $Z_{2}$ is a two-sheeted covering; [F6] shows its components are the orbits of the two-element deck action on the components upstairs. Each orbit has at most two members, so an infinite component set remains infinite after quotient; this instance is also the $r = 3, N = 2$ case of [F3].

Integer components of the based S3 gauge group

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources