Classification of Covering spaces

13 Classification of Covering spaces

13.1 Convention

13.1.1 The statement that p : E->B is a covering map will include the assumption that E and B are locally path connected and path connected.

13.1.2 We assume the general lifting correspondence theorem (Thm 54.6).

13.2 Equivalence of Covering spaces

13.2.1 Let p : E->B and p’ : E’->B be covering maps. They are said to be equivalent if there exists a homeomorphism h : E-> E’ s.t. p = p’ \bullet h. The homeomorphism h is called an equivalence of covering maps or an equivalence of covering spaces.

13.2.2 (General Lifting lemma) (Lem 79.1) Let p : E->B be a covering map; let p(e_{0}) = b_{0}. Let f : Y->B be a continuous map, with f(y_{0}) = b_{0}. Suppose Y is path connected and locally path connected. The map f can be lifted to a map \bar{f} : Y -> E s.t. \bar{f}(y_{0}) = e_{0} iff f_{}(pi_{1}(Y,y_{0})) \subset p_{}(pi_{1}(E,e_{0})). Furthermore, if such a lifting exists, it is unique.

13.2.3 (Thm 79.2) Let p : E->B and p’ : E’->B be covering maps; let p(e_{0}) = p’(e’{0}) = b{0}. There is an equivalence h : E->E’ s.t. h(e_{0}) = e’{0} iff the groups H{0} = p_{}(pi_{1}(E,e_{0})) and H’{0} = p’{}(pi_{1}(E’,e’_{0})) are equal. If h exists, it is unique.

13.2.4 If H_{1} and H_{2} are subgroups of a group G, they are said to be conjugate subgroups if H_{2} = \alpha \cdot H_{1} \cdot \alpha^{-1} for some element \alpha of G.

13.2.4.1 Conjugacy is an equivalence relation on the collection of subgroups of G.

13.2.4.2 Equivalence class of the subgroup H is called the conjugacy class of H.

13.2.5 (Lem 79.3) Let p:E->B be a covering map. Let e_{0} and e_{1} be points of p^{-1}(b_{0}) and let H_{i} = p_{*}(pi_{1}(E,e_{i})).

13.2.5.1 If \gamma is a path in E from e_{0} to e_{1}, and \alpha is the loop p \bullet \gamma in B, then the equation [\alpha] * H_{1} * [\alpha]^{-1} = H_{0} holds ; hence H_{0} and H_{1} are conjugate.

13.2.5.2 Conversely, given e_{0}, and given a subgroup H of \pi_{1}(B,b_{0}) conjugate to H_{0}, there exists a point e_{1} of p^{-1}(b_{0}) s.t. H_{1} = H.

13.2.6 (Thm 79.4) Let p:E->B and p’ : E’ -> B be covering maps; let p(e_{0}) = p’(e’{0}) = b{0}. The covering maps p and p’ are equivalent iff the subgroups H_{0} = p_{}(pi_{1}(E,e_{0})) and H’{0} = p’{}(pi_{1}(E’,e’{0})) of \pi{1} (B, b_{0}) are conjugate.

13.3 Universal covering space

13.3.1 Suppose p : E->B is a covering map, with p(e_{0}) = b_{0}. If E is simply connected, then E is called a universal covering space of B.

13.3.2 (Lem 80.1) Let B be path connected and locally path connected. Let p : E-> B be a covering map in the former sense (so that E is not required to be path connected). If E_{0} is a path component of E, then the map p_{0} : E_{0} -> B obtained by restricting p is a covering map.

13.3.3 (Lem 80.2) Let p, q and r be continuous maps with p = r \bullet q, as in the following diagram :

13.3.3.1 \begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { X & Y \ Z & \}; \path[-stealth] (m-1-1)edge node [above] {$q$} (m-1-2) edge node [left] {$p$} (m-2-1) (m-1-2) edge node [below] {$r$} (m-2-1);\end{tikzpicture}

13.3.3.2 If p and r are covering maps, so is q.

13.3.3.3 If p and q are covering maps, so is r.

13.3.4 (Thm 80.3) Let p : E->B be a covering map, with E simply connected. Given any covering map r : Y->B, there is a covering map q : E->Y s.t. r \bullet q = p.

13.3.4.1 \begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { E & Y \ B & \}; \path[-stealth] (m-1-1)edge node [above] {$q$} (m-1-2) edge node [left] {$p$} (m-2-1) (m-1-2) edge node [below] {$r$} (m-2-1);\end{tikzpicture}

13.3.4.2 This theorem shows why E is called a universal covering space of B; it covers every other covering space of B.

13.3.5 (Lem 80.4) Let p : E->B be a covering map; let p(e_{0}) = b_{0}. If E is simply connected, then b_{0} has a neighborhood U s.t. inclusion i : U ->B induces the trivial homomorphism i_{*} : \pi_{1}(U,b_{0}) -> \pi_{1}(B,b_{0}).

13.4 Covering Transformations

13.4.1 Given a covering map p : E->B, Consider the set of all equivalences of this covering space with itself; Such an equivalence is called a covering transformation.

13.4.1.1 Composites and inverses of covering transformations are covering transformations; so it forms a group of covering transformations and denoted \mathcal{C}(E,p, B).

13.4.2 Assume p : E->B is a covering map with p(e_{0}) = b_{0}; and Let H_{0} = p_{*}(\pi_{1}(E,e_{0})) throughout this section.

13.4.3 If H is a subgroup of a group G, then the normalizer of H in G is the subset of G defined by the equation N(H) = {g | gHg^{-1} = H}.

13.4.3.1 N(H) is a subgroup of G.

13.4.4 Given p : E->B with p(e_{0}) = b_{0}, let F be the set F = p^{-1}(e_{0}). Let \Phi : \ pi_{1}(B,b_{0})/H_{0} -> F be the lifting correspondence of (Thm 54.6); it is a bijection. Define also a correspondence \Psi : \mathcal{C} (E,p,B) ->F by setting \Psi (h) = h(e_{0}) for each covering transformation h : E->E.

13.4.4.1 since h is uniquely determined once its value at e_{0} is known, the correspondence \Psi is injective.

13.4.5 (Lem 81.1) The image of the map \Psi equals the image under \Phi of the subgroup N(H_{0}) / H_{0} of \pi_{1}(B,b_{0}) / H_{0}.

13.4.6 (Thm 81.2) The bijection \Phi ^{-1} \bullet \Psi : \mathcal{C} (E,p,B) -> N(H_{0})/H_{0} is an isomorphism of groups.

13.4.6.1 (Cor 81.3) The group H_{0} is a normal subgroup of \pi_{1}(B,b_{0}) iff for every pair of points e_{1} and e_{2} of p^{-1}(b_{0}), there is a covering transformation h : E->E with h(e_{1}) = e_{2}. In this case, there is an isomorphism \Phi ^{-1} \bullet \Psi : \mathcal{C} (E,p,B) -> \pi_{1}(B,b_{0}) /H_{0}.

13.4.6.2 (Cor 81.4) Let p : E->B be a covering map. If E is simply connected, then \mathcal{C} (E,p,B) \cong \pi_{1}(B,b_{0}).

13.4.7 If H_{0} is a normal subgroup of \pi_{1}(B,b_{0}), then p : E->B is called a regular covering map. (not related to ‘regular’)

13.4.8 Let X be a space, and let G be a subgroup of the group of homeomorphisms of X with itself. The orbit space X/G is defined to be the quotient space obtained from X by means of the equivalence relation x ~ g(x) for all x \in X and all g \in G. The equivalence class of x is called the orbit of x.

13.4.9 If G is a group of homeomorphisms of X, the action of G on X is said to be properly discontinuous if for every x \in X there is a neighborhood U o fx s.t. g(U) is disjoint from U whenever g \neq e. (Here e is the identity element of G) It follows that g_{0}(U) and g_{1}(U) are disjoint whenever g_{0} \neq g_{1}, for otherwise U and g^{-1}{0} g{1}(U) would not be disjoint.

13.4.10 (Thm 81.5) If p : X->B is a regular covering map and G is its group of covering transformations, then there is a homeomorphism k : X/G -> B s.t. p = k \bullet \pi, where \pi : X->X/G is the projection.

13.4.10.1 \begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { X & \ X/G & B \}; \path[-stealth] (m-1-1)edge node [above] {$p$} (m-2-2) edge node [left] {$\pi$} (m-2-1) (m-2-1) edge node [below] {$k$} (m-2-2);\end{tikzpicture}

13.5 Existence of Covering spaces

13.5.1 A space B is said to be semilocally simply connected if for each b \in B, there is a neighborhood U of b s.t. the homomorphism i_{*} : \pi_{1}(U,b) -> \pi_{1}(B,b) induced by inclusion is trivial.

13.5.2 (Thm 82.1) Let B be path connected, locally path connected, and semilocally simply connected. Let b_{0} \in B. Given a subgroup H of \pi_{1}(B,b_{0}), there exists a covering map p : E->B and a point e_{0} \in p^{-1}(b_{0}) s.t. p_{*}(\pi_{1}(E,e_{0})) = H.

13.5.2.1 (Cor 82.2) The space B has a universal covering space iff B is path connected, locally path connected, and semilocally simply connected.