TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups

Statement

Let $B$ be nonempty, path-connected, locally path-connected, and semilocally simply connected, fix $b_{0} \in B$ , and put $G = π_{1} (B, b_{0})$ .

The assignment $[p : (E, e_{0}) \to (B, b_{0})] ⟼ p_{*} π_{1} (E, e_{0})$ is a bijection from based-isomorphism classes of based connected coverings of $B$ to subgroups of $G$ .
After forgetting the chosen point in the fibre, the assignment to the conjugacy class of $p_{*} π_{1} (E, e_{0})$ is a bijection from isomorphism classes of connected coverings of $B$ to conjugacy classes of subgroups of $G$ .

Facts & Assumptions

Given: The base space and group $G$ in the Statement.

[L1]

Every subgroup $H \leq G$ is realized as the induced subgroup of a based connected quotient covering of a universal cover (Every subgroup acts on the universal cover with a connected quotient covering that realizes it).

[L2]

Over a path-connected locally path-connected base, based coverings with connected total spaces are isomorphic exactly when their induced subgroups are equal (Based connected coverings are isomorphic exactly when their induced subgroups are equal).

[L3]

For a covering with path-connected total space, changing the chosen point over $b_{0}$ conjugates the induced subgroup, and every fibre point is obtained by a lifted loop (Changing the point over a fixed basepoint conjugates the induced covering subgroup).

[F1]

Local path-connectedness lifts from the base of a covering to its total space (Local path-connectedness lifts and descends along covering maps).

[F2]

A connected locally path-connected space is path-connected (A connected, locally path-connected space is path-connected, because its path components are open).

[F3]

Every path in the base of a covering has a unique lift from a prescribed point of the fibre (Existence and uniqueness of path lifts through a covering map).

Proof

technique · direct

1.1givenL1F1F2

Every connected covering under consideration has locally path-connected total space by [F1], because $B$ is locally path-connected, and therefore has path-connected total space by [F2]. For the based correspondence, [L1] proves surjectivity: every subgroup occurs.

2.1step 1.1L2

For the based correspondence, [L2] applies under the base hypotheses and the path-connectedness established in step 1.1, and proves injectivity: two based connected coverings determine the same subgroup exactly when they are based-isomorphic. Thus claim 1 is a bijection.

2.2step 1.1L3

For claim 2, the path-connectedness from step 1.1 licenses [L3], which shows that changing the chosen point over $b_{0}$ replaces the subgroup by a conjugate. Hence the conjugacy class depends only on the unbased covering. Every conjugacy class occurs by step 1.1.

3.1step 2.1L2L3F3∎

Suppose two unbased connected coverings determine the same conjugacy class. Choose fibre points with induced subgroups $H_{1}, H_{2}$ , and write $H_{1} = g^{- 1} H_{2} g$ . By [F3], lift a loop representing $g$ from the second fibre point. By [L3], its endpoint gives a new fibre point whose induced subgroup is $H_{1}$ ; [L2] then gives a based isomorphism and hence an unbased isomorphism. Conversely, any unbased isomorphism carries a chosen fibre point to a fibre point of the other cover, so [L2] and [L3] make the subgroups conjugate. This proves injectivity and completes claim 2.

Depends on

Used by

Dependency tree · two levels

45 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, Theorem 1.38 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Chapter 3, Section 7 (standard reference, not scraped)