PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

On a connected covering space, a deck transformation is determined by one point and the deck action is free

Statement

For a covering with connected total space, two deck transformations agreeing at one point are equal. Consequently the deck group acts freely on the total space.

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

For a covering $p : E \to B$ , a deck transformation is an isomorphism $h : E \to E$ over $B$ , so $p \circ h = p$ (def-map-and-isomorphism-of-covering-spaces). Deck transformations form the deck group $Deck (p)$ under composition, and this group acts on $E$ by evaluation (def-group, def-group-action). (Deck transformations and the deck-transformation group of a covering).

[F2]

Let $Y$ be connected and let $f, g : Y \to E$ be lifts through the same covering of the same map $Y \to B$ . If $f (y_{0}) = g (y_{0})$ for some $y_{0} \in Y$ , then $f = g$ . (Two lifts from a connected space that agree at one point agree everywhere).

[F3]

A left action of a group $G$ on a set $X$ (def-group-action) is free when $g \cdot x = x ⟹ g = e$ for every $g \in G$ and $x \in X$ . Equivalently, no nonidentity element of $G$ fixes any point of $X$ . (A free group action has no nonidentity element fixing a point).

Proof

technique · direct

1.1

Two deck transformations are lifts of the same projection.

givenF1F2

2.1

If they agree at one point, uniqueness of lifts from the connected total space makes them equal.

step 1.1F2F3

3.1

Applying this to a deck transformation and the identity shows that a fixed point forces the transformation to be the identity.

step 2.1F1F3

4.1

The preceding construction and implications establish the assertion.

step 3.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Allen Hatcher, Algebraic Topology, §1.3 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Ch. 3 (standard reference, not scraped)
Marco Gualtieri, MAT1300 Week 4 Term 2, §1.6 (standard reference, not scraped)