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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

1.1

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.

givenL1F1F2

2.1

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.

step 1.1L2

2.2

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.

step 1.1L3

3.1

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.

step 2.1L2L3F3∎

Depends on

Used by

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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)