A covering space of a topological space \(X\) is a space \(X̃\) together with a continuous function \(p: X̃ \to X\) satisfying: There is an open covering \(\{U\alpha \}\) of \(X\) such that for all \(\alpha\) the preimage \(p\inv(U\alpha )\) is a disjoint union of open sets in \(X̃\text{,}\) each of which is mapped (by the restriction of \(p\)) homeomorphically onto \(U\alpha\text{.}\) Each set \(U\alpha\) is called evenly covered.
Example6.1.2.
Examples
Subsection6.1.2Lifting Theorems
Theorem6.1.3.PPHLT = Path and Path Homotopy Lifting Theorems.
Let \(p: (X̃,x̃_0) \to (X,x_0)\) be a covering space.
PLT.
Given a path \(f: (I,0) \to (X,x_0)\text{,}\) there is a unique path \(f̃: (I,0) \to (X̃,x̃_0)\) such that \(p \circ f̃ = f\text{.}\)
PHLT.
Given a path homotopy \(H: (I \times I,(0,0)) \to (X,x_0)\text{,}\) there is a unique path homotopy \(H̃: (I \times I,(0,0)) \to (X̃,x̃_0)\) such that\(p \circ H̃ = H\text{.}\) Moreover, if \(H\) is a path homotopy between two paths \(f,g: (I,0) \to (X,x_0)\text{,}\) then \(H̃\) is a path homotopy between the "lifts" \(f̃,g̃: (I,0) \to (X̃,x̃_0)\) of \(f,g\) from the PLT.
Theorem6.1.4.LC+ULP = Lifting Criterion and Unique Lifting Property.
Let \(p: (X̃,x̃_0) \to (X,x_0)\) be a covering space, let \(Y\) be a PC and LPC space, and let \(f: (Y,y_0) \to (X,x_0)\) be continuous.
If \(f∗(\pi_1(Y,y_0)) \sse p∗(\pi_1(X̃,x̃_0))\text{,}\) then there is a unique continuous function \(f̃: (Y,y_0) \to (X̃,x̃_0)\) such that \(p \circ f̃ = f\text{.}\)
If \(f∗(\pi_1(Y,y_0)) ⊈ p∗(\pi_1(X̃,x̃_0))\text{,}\) then there does not exist a continuous lift \(f̃: (Y,y_0) \to (X̃,x̃_0)\) such that \(p \circ f̃ = f\text{.}\)
Theorem6.1.5.
If \(p: (X̃,x̃_0) \to (X,x_0)\) is a covering space, then the induced homomorphism \(p∗: \pi_1(X̃,x̃_0) \to \pi_1(X,x_0)\) is injective (that is, \(Ker p∗ =\{1\pi_1(X̃,x̃_0)\}\)), and the image satisfies \(Im p∗ = p∗(\pi_1(X̃,x̃_0)) =\{[f] \in \pi_1(X,x_0) | \text{ the lift } f̃ \text{ of }f \text{ at }x̃_0 \text{ is a loop in }X̃\}\text{.}\) Consequently, \(\pi_1(X̃,x̃_0)\) is isomorphic to a subgroup of \(\pi_1(X,x_0)\text{.}\)
Subsection6.1.3Application to group theory
Corollary6.1.6.
For all \(n \in \N\text{,}\) the free group \(Fn\) is a subgroup of \(F2\text{.}\)
The free group \(F(S)\) on a countably infinite set \(S\) is a subgroup of \(F2\text{.}\)
Subsection6.1.4The number of sheets
Definition6.1.7.
Let \(p: (X̃,x̃_0) \to (X,x_0)\) be a covering space such that \(X\) and \(X̃\) are PC. The number of sheets of the covering space is \(|p\inv({x_0\})|\text{.}\)
Theorem6.1.8.Lifting Correspondence.
Let \(p: (X̃,x̃_0) \to (X,x_0)\) be a covering space such that \(X\) and \(X̃\) are PC. Then:
The Lifting Correspondence Function \(Φ: \pi_1(X,x_0)/p∗(\pi_1(X̃,x̃_0)) \to p\inv({x_0\})\) defined by \(Φ(p∗(\pi_1(X̃,x̃_0))[f]) := f̃(1)\text{,}\) where \(f̃\) is the unique lift of the loop \(f\) to a path in \(X̃\) starting at \(x̃_0\text{,}\) is a well-defined bijection.
For all points \(x' \in X, |p\inv({x'\})| = |p\inv({x_0\})| = \)(the number of sheets of\(p\)) \(=\) the index of \(p∗(\pi_1(X̃,x̃_0)) in \pi_1(X,x_0)\text{.}\)
Example6.1.9.
Examples
Subsection6.1.5Interactions with functions and constructions:
Theorem6.1.10.
If \(p:X̃ \to X\) is a covering space, then \(p\) is an open map.
Theorem6.1.11.
Let \(p:X̃ \to X\) be a covering space. If \(A\) is a subspace of \(X\) and \(Ã := p\inv(A)\text{,}\) then the restriction \(p|AÃ:Ã \to A\) is also a covering space.
Theorem6.1.12.
Let \(p_i:X̃_i \to X_i\) be covering spaces for \(i \in\{1,2\}\text{.}\) Then \(q := (p_1 \circ proj_1,p_2 \circ proj_2) :X̃_1 \times X̃_2 \to X_1 \times X_2 [\text{ that is}, q(S^1,S^2) := (p_1(S^1),p2(S^2)) \text{ for all }(S^1,S^2) \in X̃_1 \times X̃_2]\) is also a covering space.
Subsection6.1.6Interactions with homeomorphism invariants:
Corollary6.1.13.
Let \(p:X̃ \to X\) be a covering space. If \(X\) is a CW complex, then \(X̃\) is a CW complex. Moreover, \(p\) maps open \(n\)-cells to open \(n\)-cells for all \(n\text{.}\)
Theorem6.1.14.
Let \(p:X̃ \to X\) be a covering space satisfying the property that \(p\inv({s\})\) is finite and nonempty for all \(s \in X\text{.}\) The space \(X̃\) is compact and \(T_2\) if and only if the space \(X\) is compact and \(T_2\text{.}\)
