PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.

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Equivariant maps descend on free proper quotients

Statement

Let $M$ and $N$ be smooth free proper $G$ -manifolds. Every smooth $G$ -equivariant map $f : M \to N$ induces a unique smooth map $\overline{f} : M / G \to N / G$ such that

$\overline{f} \circ q_{M} = q_{N} \circ f .$

Facts & Assumptions

Given: The two free proper $G$ -manifolds, their quotient maps, and a smooth equivariant map $f : M \to N$ .

[F1]

Equivariance means $f (g \cdot x) = g \cdot f (x)$ . Equivariant maps and equivariant vector bundles.

[F2]

The quotient maps are smooth surjective submersions. Free proper action quotient manifold.

[F3]

A continuous map constant on quotient fibres factors uniquely and continuously through the quotient. For a quotient map $q : X \to Y$ , a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$ ; and a composite of quotient maps is a quotient map.

[F4]

A smooth submersion has local projection form and smooth local sections. The constant-rank theorem for manifolds.

Proof

Proof technique: quotient universality followed by local submersion sections.

1.1

If $q_{M} (x) = q_{M} (y)$ , then $y = g \cdot x$ for some $g \in G$ . By [F1], $f (y) = g \cdot f (x)$ , so $q_{N} (f (y)) = q_{N} (f (x))$ . Thus the smooth map $q_{N} \circ f$ is constant on the fibres of $q_{M}$ .

F1given

2.1

Apply [F3] to obtain a unique continuous map $\overline{f} : M / G \to N / G$ with $\overline{f} \circ q_{M} = q_{N} \circ f$ .

F2F3step 1.1

3.1

Let $u \in M / G$ . Since $q_{M}$ is a submersion by [F2], [F4] supplies a smooth local section $s : U \to M$ near $u$ . On $U$ , the factorization identity gives $\overline{f} ∣_{U} = q_{N} \circ f \circ s$ , which is smooth. Such neighborhoods cover $M / G$ , so $\overline{f}$ is smooth. Uniqueness as a smooth map follows from the uniqueness in [F3]. No choice principle is used.

F2F3F4step 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)