TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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$Deck (E / B) ≅ N_{G} (H) / H$ for a connected covering

Statement

Let $p : (E, e_{0}) \to (B, b_{0})$ be a connected covering of a path-connected locally path-connected base. Put $G = π_{1} (B, b_{0})$ and $H = p_{*} π_{1} (E, e_{0})$ . Then

$Deck (E / B) ≅ N_{G} (H) / H .$

Facts & Assumptions

Given: The covering and subgroups $H \leq N_{G} (H) \leq G$ in the Statement.

[L1]

There is a surjective homomorphism $Θ : N_{G} (H) \to Deck (E / B)$ with kernel $H$ (Deck transformations of a connected covering correspond to cosets in the subgroup normalizer).

[F1]

The first isomorphism theorem gives $K / \ker f ≅ im f$ for a group homomorphism $f : K \to L$ (First isomorphism theorem for groups: $G / \ker f ≅ im f$ ).

[F2]

An element of $N_{G} (H)$ conjugates $H$ to itself (The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup).

[F3]

The normalizer $N_{G} (H)$ is a subgroup of $G$ ( $C_{G} (x)$ and $N_{G} (H)$ are subgroups of $G$ ).

Proof

technique · direct

1.1

Use [L1] to take the surjective homomorphism $Θ : N_{G} (H) \to Deck (E / B)$ .

1.2

By [F3], $N_{G} (H)$ is a group. By [F2], $n H n^{- 1} = H$ for every $n \in N_{G} (H)$ , so $H ⊴ N_{G} (H)$ and the quotient $N_{G} (H) / H$ is defined. By [L1], $\ker Θ = H$ .

L1F2F3

2.1

Applying [F1] to $Θ$ gives $N_{G} (H) / H = N_{G} (H) / \ker Θ ≅ im Θ = Deck (E / B) .$

step 1.1step 1.2F1∎

Depends on

Used by

A regular connected covering has deck group π₁(B,b₀)/p_*π₁(E,e₀) Corollary

Dependency tree · two levels

25 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

Allen Hatcher, Algebraic Topology, Proposition 1.39(b) (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Chapter 3, Section 7 (standard reference, not scraped)

Deck⁡(E/B)≅NG(H)/HDeck(E/B)≅NG​(H)/H for a connected covering