Two perspectives on the classification of covering spaces Saul Glasman December 14, 2016 Proof. Continuation of the proof of the existence of a the universal cover for a locally path-connected, semilocally simply connected spaces: It remains to show that E is simply connected. First we'll show it's path- connected. Indeed, let α be a path from b0 to b, and for each s 2 [0; 1], let α(s) : [0; 1] ! B be defined by α(s)(t) = α(st): (s) So α travels along α from b0 to α(s) at constant speed. Then the path α0 : [0; 1] ! E; α0(t) = [α(t)] can be proved to be continuous, and it's a path from the constant path at b0 to [α] in E. Finally, we need to show that pi1(E; e0) is trivial. We previously showed that if p is a covering map, then p∗ : π1(E; e0) ! pi1(B; b0) is injective. That means if [f] is a nontrivial loop in E, then [p ◦ f] is a nontrivial loop in B, and moreover, φ([p ◦ f]) = e0 where φ is the lifting correspondence associated to p. This can be stated as a theorem: −1 Theorem 1. The subgroup φ (e0) of π1(B; b0) contains the image of p∗. −1 In particular, if we can show that φ (e0) is the identity subgroup, this will imply that im (p∗) is trivial, and since p∗ is injective, this means that π1(E; e0) itself is trivial. But if [f] 2 π1(B; b0), then [f] lifts to a path from [cb0 ] to [f] 2 E, by the same method we used to show path-connectedness of E. If [f] is a nontrivial element of π1(B; b0), then [f] 6= [cb0 ] 2 E by definition. This proves the theorem. The final step is to classify covering spaces, and I'm going to just briefly sketch the terrain here. The classification can be carried out in two different ways: one algebraic and one topological. 1 Definition 2. Let G be a group and let H0;H1 be two subgroups of G. H0 −1 and H1 are said to be conjugate if there is some g 2 G such that g H0g = H1: −1 that is, h 2 H0 if and only if g hg 2 H1. For instance, a normal subgroup is only conjugate to itself. Conjugacy of subgroups is an equivalence relation, and an equivalence class is called a conju- gacy class. Definition 3. Let B be a space and let p0 : E0 ! B and p1 : E1 ! B be cover- ing maps. Then we say E0 and E1 are equivalent if there is a homeomorphism h : E0 ! E1 such that the diagram h E0 E1 p0 p1 B commutes. Equivalence of covering spaces is again an equivalence relation. Here's a theorem classifying covering spaces: Theorem 4. Let B be a path-connected, locally path-connected, semilocally simply connected space. Then there is a bijection between the set of equivalence classes of path-connected covering spaces of B and the set of conjugacy classes of subgroups of π1(B; −). To prove this, we need a theorem that vastly generalizes path lifting and path homotopy lifting. Theorem 5. Let p : E ! B be a covering map. Let e0 2 E and p(e0) = b0. Suppose (Y; y0) is another pointed spaces, which is path-connected and locally path-connected, and let f : Y ! B be a map with f(y0) = b0. Then f has a lift f along p with f(y0) = e0 if and only if f∗(π1(Y; y0)) ⊆ p∗(π1(E; e0)): If it exists, the lifting is unique. Note that if Y is simply connected, the condition is always satisfied. This gives path lifting and path homotopy lifting. (But the proof uses path lifting and path homotopy lifting.) Theorem 6. Let p0 : E0 ! B and p1 : E1 ! B be covering maps, and e0 2 E0; e1 2 E1 with p0(e0) = p1(e1) = b0. Then E0 and E1 are equivalent covering spaces if and only if H0 = p0∗(π1(E0; e0)) is conjugate to H1 = p1∗(π1(E1; e1)). Proof. The point is that changing the basepoint conjugates the subgroup: if 0 −1 0 e1 2 p1 (b0), then H1 is conjugate to H1 = p1∗(π1(E1; e1)). Conversely, if 0 −1 0 H0 and H1 are conjugate, we can find some e1 2 p1 (b0) such that H1 = H0. Then the lifting theorem shows that p0 has a unique lift p0 along p1 such that 0 p0(e0) = e1, and it can be shown that p0 is a homeomorphism. 2 Theorem 7. Suppose H ⊆ π1(B; b0) is a subgroup. Then there is a covering map p : E ! B such that p∗(π1(E; e0)) = H. Proof. I won't say anything about this, except to say that constructing the universal cover (corresponding to H = feg) was most of the work - in general, we can construct E as a quotient space of the universal cover. This completes a very sketchy proof of the algebraic classification of covering spaces. Now for the topological classification: Definition 8. Let G be a group. A space X is said to be a classifying space for the group G if • X satisfies some \niceness" (which I won't go into, but it's along the lines of being locally contractible), ∼ • π1(X; −) = G, and • the universal cover of X is contractible. Theorem 9. Any two classifying spaces for G are homotopy equivalent. If we're doing homotopy theory, we will often speak of \the" classifying space of G and call it BG. Theorem 10. Let B be a (sufficiently nice) space. Then there is a bijection between [B; BSn], the set of homotopy classes of maps from B to the classifying space of the symmetric group Sn, and the set of equivalence classes of (not necessarily path-connected) degree n covers of B. 3.