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Hatcher §1.3 Hatcher §1.3 Ex 1.3.7 The quasi-circle W ⊂ R2 is a compactification of R with remainder W − R = [−1, 1]. There is a quotient map q : W → S1 to the one-point compactification S1 of R obtained by collapsing [−1, 1] to a point. This map is manifestly continuous (but there is also a general reason [2]). We first show that any map from a locally path-connected compact space to W factors through a contractible subspace of W . Let X be any locally path-connected compact space and f : X → W a continuous map. There is an open set V ⊂ W containing [−1, 1] with two path-components, V+ ⊃ [−1, 1] and V−. If x is a point of X such that f(x) ∈ [−1, 1] then there is a path-connected neighborhood of x that is being −1 mapped into V+. Thus f [−1, 1] is contained in an open set U such that f(U) ⊂ V+. Now X − U is a compact subspace of X that is being mapped into the remainder R ⊂ W . By compactness, f(X − U) ⊂ [a, b] for some closed interval [a, b] ⊂ R. Thus the f(X) = f(U) ∪ f(X − U) is contained in U+ ∪ [a, b] which again is contained in a compact subspace of W homeomorphic to [−1, 1] ∨ [0, 1]. In particular, π1(W ) = 0 (and, actually, πn(W ) = 0 for all n ≥ 1). Suppose that f is a lift R > f | || p || || | 1 W q / S of the quotient map q. Then f([−1, 1]) ⊂ Z and we may as well assume that f([−1, 1]) = {0}. The restriction of f to R = (0, 1) ⊂ W has the form f(t) = t + n for some n ∈ Z by uniqueness of lifts on connected spaces. But this contradicts continuity of f for there is no neighborhood U of 1 1 0 ∈ [−1, 1] such that f(U) ⊂ (− 2 , 2 ). (Any neighborhood U of 0 ∈ [−1, 1] ⊂ W contains points with f-values near n and also points with f-values near n+1 in that f(U ∩(0, 1)) = (U ∩(0, 1))+n.) Ex 1.3.9 1 Since π1(X) is finite, the induced homomorphism f∗ : π1(X) → π1(S ) = Z is trivial as there are no nontrivial finite subgroups of Z. By the Lifting criterion [1, 1.33], f : X → S1 factors through the universal covering space R of S1, f = pf˜ where p: R → S1 is the universal covering map. But R is contractible, the identity map of R is homotopic to the constant map, so any map into R is homotopic to constant map. In particular, f˜ ' ∗, so also f = pf˜ ' p∗ = ∗. 1.3.12 1 1 2 2 4 Let N ⊂ G = ha, bi = π1(S ∨S ) be the smallest normal subgroup containing a , b , and (ab) . 1 1 The universal covering space of S ∨ S is the Cayley graph XeG of the right G-set G = {e}\G [1, Example 1.45]. The covering space of S1 ∨ S1 corresponding to the subgroup N < G is the orbit space N\XeG for the N-action on XeG which is equivalent [3] to the Cayley graph 1 1 XeN\G = N\XeG → G\XeG = XeG\G = S ∨ S 1 2 a for the right G-set N\G. This Cayley graph XeN\G is just the set N\G with edges x −→ xa and x −→b xb for all x ∈ N\G. N(ab)2 7 h a b a b v ( 6 Nbab Naba h b a b a Nba u ) Nab i K a b a b ) Nb Na j I b a b a Ù * Ne Ex 1.3.14 (The Cayley complex for the infinite dihedral group G = a, b|a2b2 .) Let G be the group G = a, b|a2b2 = Z/2 ∗ Z/2. The subgroup generated by ab is an infinite cyclic normal subgroup with complement hai = Z/2 so that G = Z o Z/2 is a semi-direct product. The group Z/2 = hai acts on Z = habi by taking ab to aaba = ba = (ab)−1. The Cayley complex for the group G, a 0 a 1 a 2 XeG = D ∪ D ∪ D G G×{a,b} G×{a2,b2} an infinite string (infinite in both directions) of spheres holding hands [1, Example 1.48], is the universal covering space of 0 a 1 a 2 2 2 XG = G\XeG = D ∪ D ∪ D = RP ∨ RP , {a,b} {a2,b2} the wedge of two copies of the real projective plane. The universal covering space action G×XeG → XeG is such that • ab translates the infinite string two places to the right, • a flips the whole string around the S2 with hands e and a and acts as the antipodal map on this S2. Since a(ab)a = ba = (ab)−1, the subgroup habi generated by ab is normal. The quotient group has order two and there is a short split exact sequence of the form 0 → Z → G → Z/2 −→` 0 where the homomorphism ` records word length parity. Thus the subgroups of G = Z o Z/2 are n n Hn = h(ab) i and Kn = h(ab) , ai for some n ≥ 0. Let us imagine that the sphere S2 has two hands placed at opposite points. The quotient space RP 2 then has only one hand. Thus RP 2 ∨ RP 2 is two RP 2s in love and holding hands. The covering spaces of RP 2 ∨ RP 2 consist of S2s and RP 2s holding hands such that there are no free hands. The functor OG → Cov(G\XeG): H\G → H\XeG = XeH\G is an equivalence of categories. We must figure out what these quotients, the Cayley complexes for the orbit spaces H\G, are. 3 The group Hn has index 2n and the Cayley complex for Hn\G [3] is the space Xn consisting of 2n S2s holding hands in a circle, a necklace with 2n pearls. The group Kn has index n and the Cayley complex for Kn\G is the space Yn consisting of a string of n − 1 spheres holding hands and holding a P 2 at each end. 2 2 Let Y∞ consist of a string of S holding hands, infinite in one direction, and holding a RP at one end. The image of π1(Y∞) = Z/2 is the subgroup hai. What is the deck transformation group? The space Xn admits a Z/2 covering space action reflecting the whole arrangement across a line 2 through two S s opposite each other. The quotient space is Yn, 1 ≤ n ≤ ∞ [3, 2.6]. 2 2 Since there are no more subgroups of π1(RP ∨ RP ), we have described all covering spaces of RP 2 ∨ RP 2. Ex 1.3.18 The space X has a universal covering space action G × Y → Y where G = π1(X, x) is the fundamental group and Y is the universal covering space. Any covering map over X = G\Y is isomorphic to a covering map of the form pGH : H\Y → G\Y for some subgroup H < G. The Noether isomorphism theorems [4, 1.4] in group theory tell us that 0 pGH : H\Y → G\Y is abelian ⇔ H is normal in G and G/H is abelian ⇔ G < H where G0 = [G, G] is the derived subgroup generated by all commutators. This means that the 0 covering space pGG0 : G\Y → G \Y corresponding to the derived subgroup is an abelian covering map which is universal in the sense that it lies above any other abelian covering space of X because H\Y = (G0\H)\(G0\Y ). Its deck transformation group is (G0\H)\(G0\G). 0 Write Gab = G/G for the largest abelian quotient of G. We may characterize the universal abelian covering map by either of these two equivalent properties: • The right G-set corresponding to the universal abelian covering map is isomorphic to Gab (which is a right G-set by the standard action Gab × G → Gab). • The universal abelian covering map is the normal covering map with Gab as deck trans- formation group. Suppose that X = S1 ∨ S1 is a wedge of two circles. Let X0 = Z × R ∪ R × Z ⊂ R2 be the space of all integer grid lines in R2. There is an obvious covering map p: X0 → X. The fibre, −1 the integer grid points p (∗) = Z × Z, is isomorphic to the right Z ∗ Z-set (Z ∗ Z)ab = Z × Z. Therefore p: X0 → X is the universal abelian covering map over X = S1 ∨ S1. The covering space (mZ × nZ)\X0 (a graph that can be drawn on a torus (mZ × nZ)\(R × R) is an abelian covering space of S1 ∨ S1 with Z/m × Z/n as its deck transformation group. It is easy to generalize this example to X = S1 ∨ ... ∨ S1. Ex 1.3.20 (The Cayley complex for G = a, b|abab−1 ) −1 1 1 2 Consider the group G = a, b|abab . Then XG = (S ∨ S ) ∪abab−1 D = N2 is the nonori- −1 entable compact surface N2 of genus 2, the Klein Bottle. From the relation ab = ba we see that ab2 = aab = aba−1 = ba−1a−1 = b2a, i.e. that a and b2 commute. The Cayley complex for 2 G, the universal covering space of N2, is the plane R and the universal covering space action [1, Example 1.42] G × R2 → R2, a(x, y) = (x + 1, y) is horizontal translation by one unit and b(x, y) = (0, 1)+(−x, y) is the glide-reflection consisting of vertical translation by one unit followed by reflection across the y-axis.
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