LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingjudge pass (gpt-6-luna)

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.

Evaluation and finite-center fibrations of sphere gauge groups

Statement

Let $P \to S^{r}$ be a smooth principal $S U (N)$ bundle with $r \in {3, 4}$ , $N \geq 2$ , a chosen point $m \in S^{r}$ , and a chosen frame $p_{m}$ in its fiber. Write $G (P) : = G_{\infty} (P)$ for the smooth gauge group defined by the finite-atlas $C^{\infty}$ section topology, let $G_{m} (P) : = {u \in G_{\infty} (P) : u (p_{m}) = e}$ , and put $\overline{G} (P) : = {\overline{G}}_{\infty} (P) = G_{\infty} (P) / Z_{N}$ , where $Z_{N}$ is the finite subgroup of constant central gauge sections. Evaluation $e v_{m} (u) = u (p_{m})$ gives a locally trivial bundle $G_{m} (P) ⟶ G (P) \overset{e v_{m}}{\to} S U (N),$ and the quotient map $G (P) \to \overline{G} (P)$ is a finite covering. For a local section $s : U \to G (P)$ , the bundle chart has the commutative form $\begin{array}{ccc} G_{m} (P) \times U & \overset{Φ_{s}}{\to} & e v_{m}^{- 1} (U) \\ ↓ {pr}_{2} & ↓ e v_{m} \\ U & ↪ & S U (N) \end{array}$ where $Φ_{s} (v, g) = v s (g)$ ; the square commutes because $e v_{m} (v) = e$ and $e v_{m} (s (g)) = g$ . The corresponding TikZ source is:

The evaluation bundle is numerable; both maps are Serre fibrations and have the homotopy long exact sequence.

Statement references

May, A Concise Course in Algebraic Topology, Chapters 7 and 9

Hatcher, Algebraic Topology, Chapter 4

Facts & Assumptions

Given: The bundle and frame of the Statement. Use the actual smooth-section conventions below; their 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 actual configuration definition fixes compact connected oriented boundaryless smooth Riemannian $M$ with $d \in {3, 4}$ , a smooth principal $S U (N)$ bundle $P$ , frame $p_{m}$ , the adjoint group bundle, the smooth gauge group $G_{\infty}$ with finite-atlas $C^{\infty}$ section topology, its based subgroup, and the constant central sections $Z_{N}$ . It also defines the Sobolev spaces, $A_{\infty}$ , and ${\overline{G}}_{\infty} = G_{\infty} / Z_{N}$ . The carrier is present, its Definition has been inspected, and its hash-bound Step 1 record is current and ready; this is authoring readiness, not independent approval. Smooth and Sobolev configuration and reduced gauge spaces

[F2]

A locally trivial fiber bundle is defined by supplied local product charts; a numerable such bundle is a Hurewicz fibration under AC. Locally trivial fiber bundle, Numerable fiber bundles are hurewicz fibrations, The Axiom of Choice

[F3]

A based Serre fibration has the long exact sequence of homotopy groups and component sets. Long exact sequence of homotopy groups of a fibration

[F4]

The last-column map $S U (N) \to S^{2 N - 1}$ is locally trivial with fiber $S U (N - 1)$ . Special-unitary column fibration

[F5]

Every sphere $S^{d}$ with $d \geq 1$ is path connected. For $n \geq 2$ , the sphere $S^{n - 1}$ is path-connected and connected

[F6]

Covering maps have the homotopy lifting property, so their induced homotopy-group sequences apply. Existence and uniqueness of homotopy lifts through a covering map

[A1]

Choose a bundle chart over a neighborhood of $m$ and the frame $p_{m}$ to identify the evaluated fiber with the matrix group $S U (N)$ . The central subgroup $Z_{N}$ consists of the scalar matrices $ζ I$ with $ζ^{N} = 1$ .

Proof

1.1A1F3F4F5construct

In a trivializing ball around $m$ , choose a smooth bump $b$ supported in that ball with $b (m) = 1$ . For $g$ near $I$ , use a smooth logarithm $X = \log g \in s u (N)$ and define a gauge section by $u_{g} (x) = \exp (b (x) X)$ in that chart and the identity outside. This gives a continuous local section of evaluation near $I$ . To see that every value is attained, begin at $N = 2$ : [F4] gives a bundle $S U (1) \to S U (2) \to S^{3}$ with one-point fiber, and [F5] makes the base path connected; the component exactness [F3] therefore makes $S U (2)$ path connected. Inductively, [F4] is a surjective locally trivial bundle with path-connected fiber $S U (N - 1)$ and path-connected base sphere [F5]; path lifting and [F3] imply $S U (N)$ is path connected. A path from $I$ to any $g_{0}$ is covered by finitely many logarithm neighborhoods; subdividing it writes $g_{0}$ as a finite product of elements in the identity chart. Multiplying the corresponding bump-supported sections gives a gauge section $v_{0}$ with $v_{0} (p_{m}) = g_{0}$ . Translation of the identity-neighborhood section by $v_{0}$ yields a local section near every $g_{0}$ .

1.2A1F1given

The finite group $Z_{N}$ acts on the Hausdorff topological group $G (P)$ by free translations. Choose an identity neighborhood $V$ with $V^{- 1} V \cap Z_{N} = {e}$ ; this is possible because the finite set $Z_{N} ∖ {e}$ is closed and avoids the identity. The translates $V z$ are pairwise disjoint. Since the quotient map is open and identifies each translate homeomorphically with the image of $V$ , that image is evenly covered. Translation gives the same construction around every group element, so the quotient map is a covering with exactly $∣ Z_{N} ∣ = N$ sheets. By [F1] and [A1], this is the finite-center quotient specified in the Statement.

2.1F1F2F3step 1.1construct

In the finite-atlas $C^{\infty}$ topology, pointwise multiplication and inversion are continuous: each derivative seminorm of a product or inverse is bounded on compact chart supports by finitely many derivative seminorms of the inputs, by the finite-dimensional product and chain rules. If $s : U \to G (P)$ is one of the local sections, the maps $(v, g) ⟼ v s (g), u ⟼ (u s (e v_{m} (u))^{- 1}, e v_{m} (u))$ are continuous inverses between $G_{m} (P) \times U$ and $e v_{m}^{- 1} (U)$ . The first coordinate in the inverse evaluates to $e$ , and the product reconstructs $u$ . Thus evaluation is a locally trivial bundle with fiber exactly its kernel $G_{m} (P)$ . A finite cover of compact $S U (N)$ by such neighborhoods and a partition of unity make this bundle numerable. By [F2] it is a Hurewicz fibration; [F3] supplies its homotopy LES.

3.1F2F3F6step 2.1step 1.2given∎

The base $S U (N)$ is compact, so the finite local trivializations of evaluation admit a subordinate partition of unity. By [F2, F3] the evaluation bundle is a Serre fibration with the homotopy LES. The finite quotient is a covering; [F6] gives its homotopy lifting property and hence its fibration property. This completes the claims.

Evaluation and finite-center fibrations of sphere gauge groups

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources