Proposition 1.23 Let X be a topological space which is connected and locally path-connected, let X˜ be a connected topological space, let p: X˜ → X be a covering map over X, and let w₁ and w₂ be points of the covering space X˜.

Then there exists a deck transformation h: X˜ → X˜ sending w₁ to w₂ if and only if p_#(π₁(X, w˜₁)) = p_#(π₁(X, w˜₂)), in which case the deck transformation sending w₁ to w₂ is uniquely determined.

Proof The proposition follows immediately on applying Theorem 1.21.

Corollary 1.24 Let X be a topological space which is connected and locally path-connected, let p: X˜ → X be a covering map over X, where the covering space X˜ is connected. Suppose that p_#(π₁(X, w˜₁)) is a normal subgroup of π₁(X, p(w₁)). Then, given any points w₁ and w₂ of the covering space X˜ satisfying p(w₁) = p(w₂), there exists a unique deck transformation h: X˜ → X˜ satisfying h(w₁) = w₂.

Proof The covering space X˜ is both locally path-connected (by Proposition 1.16) and connected, and is therefore path-connected (by Corollary 1.17).

It follows that p_#(π₁(X, w˜₁)) and p_#(π₁(X, w˜₂)) are conjugate subgroups of π₁(X, p(w₁)) (Lemma 1.7). But then p_#(π₁(X, w˜₁)) = p_#(π₁(X, w˜₂)), since p_#(π₁(X, w˜₁)) is a normal subgroup of π₁(X, p(w₁)). The result now follows from Proposition 1.23.

Theorem 1.25 Let p: X˜ → X be a covering map over some topological space X which is both connected and locally path-connected, and let x₀ and x˜₀ be points of X and X˜ respectively satisfying p(x˜₀) = x₀. Suppose that X˜ is connected and that p_#(π₁(X˜, x˜₀)) is a normal subgroup of π₁(X, x₀).

Then the group Deck(X˜|X) of deck transformations is isomorphic to the corresponding quotient group π₁(X, x₀)/p_#(π₁(X˜, x˜₀)).

Proof Let G be the group Deck(X˜|X) of deck transformations of X˜. Then the group G acts freely and properly discontinuously on X˜ (Proposition 1.13).

Let q: X → X/G ˜ be the quotient map onto the orbit space X/G. Elements w₁ and w₂ of X˜ satisfy w₂ = g(w₁) for some g ∈ G if and only if p(w₁) = p(w₂).

It follows that there is a continuous map h: X/G ˜ → X for which h ◦ q = p.

This map h is a bijection. Moreover it maps open sets to open sets, for if W is some open set in X/G ˜ then q⁻¹ (W) is an open set in X˜, and therefore p(q⁻¹ (W)) is an open set in X, since any covering map maps open sets to open sets. But p(q⁻¹ (W)) = h(W). Thus h: X/G ˜ → X is a continuous bijection that maps open sets to open sets, and is therefore a homeomorphism. The fundamental group of the topological space X is thus isomorphic to that of the orbit space X/G ˜ . It follows from Proposition 1.9 that there exists a surjective homomorphism from π1(X, x₀) to the group G of deck transformations of the covering space. The kernel of this homomorphism is p_#(π₁(X˜, x˜₀)). The result then follows directly from the fact that the image of a group homomorphism is isomorphic to the quotient of the domain by the kernel of the homomorphism.

Corollary 1.26 Let p: X˜ → X be a covering map over some topological space X which is both connected and locally path-connected, and let x₀ be a point of X. Suppose that X˜ is simply-connected. Then Deck(X˜|X)≅ π₁(X, x₀).

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