Homework 3 MTH 869 Algebraic Topology

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1 Homework 3 MTH 869 Algebraic Topology Joshua Ruiter February 12, 2018 Proposition 0.1 (Exercise ). Let (X, x 0 ) and (Y, y 0 ) be pointed, path-connected spaces. Let f : I X y 0 } and g : I x 0 } Y both be loops based at (x 0, y 0 ). Via inclusions, we can think of f, g as loops I X Y based at (x 0, y 0 ). Let p X : X Y X and p Y : X Y Y be the standard projections. Then we have f g g f via the homotopy g(2s) 0 s t/2 h t (s) (p X f(2s t), p Y g(t)) t/2 s t/2 + 1/2 g(2s 1) t/2 + 1/2 s 1 As a consequence, we have [f][g] [g][f]. Proof. Define h t as above. We check that the potentially conflicting definitions agree on the overlaps. When s t/2, we have h t (s) g(2s) g(t) (x 0, p Y g(t)) h t (s) (p X f(2(t/2) t), p Y (g(t)) (p X f(0), p Y g(t)) (x 0, p Y g(t)) When s t/2 + 1/2, we have h t (s) (p X f(2(t/2 + 1/2) t), p Y g(t)) (p X f(t + 1 t), p Y g(t)) (p X f(1), p Y g(t)) (x 0, p Y g(t)) h t (s) (x 0, p Y g(2(t/2 + 1/2) 1)) (x 0, p Y g(t + 1 1)) (x 0, p Y g(t)) Now we check that h t is a homotopy of paths. It is immediate to check that it fixes the endpoints for all t: h t (0) g(0) (x 0, y 0 ) h t (1) g(2(1) 1) g(1) (x 0, y 0 ) 1

2 Now we show that h 0 f g and h 1 g f. g(2s) 0 s 0 h 0 (s) (p X f(2s), p Y g(0)) 0 s 1/2 g(2s 1) 1/2 s 1 (p X (f(2s), y 0 ) 0 s 1/2 g(2s 1) 1/2 s 1 f(2s) 0 s 1/2 g(2s 1) 1/2 s 1 f g(s) g(2s) 0 s 1/2 h 1 (s) (p X f(2s 1), p Y g(1)) 1/2 s 1 g(2s 1) 1 s 1 g(2s) 0 s 1/2 (p X (f(2s 1), y 0 ) 1/2 s 1 g(2s) 0 s 1/2 f(2s 1) 1/2 s 1 g f(s) Thus f g g f. Hence [f][g] [f g] [g f] [g][f] Proposition 0.2 (Exercise ). Let X, Y be path connected spaces. Let p 1 : X Y X and p 2 : X Y Y be the projections (x, y) x and (x, y) y respectively. We have induced homomorphisms p 1 : π 1 (X Y ) π 1 (X) and p 2 : π 1 (X y) π 1 (Y ). Define φ : π 1 (X Y ) π 1 (X) π 1 (Y ) by Then φ is a group isomorphism. [f] (p 1 [f], p 2 [f]) Proof. First we show that φ is a group homomorphism. Since p 1 is a homomorphism, and likewise for p 2. Thus p 1 ([f][g]) (p 1 [f])(p 1 [g]) φ([f][g]) (p 1 ([f][g]), p 2 ([f][g])) ( (p 1 [f])(p 1 [g]), (p 2 [f])(p 2 [g]) ) ( p 1 [f], p 2 [f] )( p 1 [g], p 2 [g] ) (φ[f])(φ[g]) 2

3 so φ is a homomorphism. Now we show that φ is surjective. Let ([f x ], [f y ]) π 1 (X) π 1 (Y ), and choose representatives f x, f y. Define f : I X Y by f(t) (f x (t), f y (t)). Then φ[f] (p 1 [f], p 2 [f]) ([p 1 f], [p 2 f]) ([f x ], [f y ]) Hence φ is surjective. Finally, we show that φ is injective by showing that the kernel is trivial. Let [f] ker φ, and choose a representative f. Since [f] ker φ, we have p 1 [f] [p 1 f] 0 and p 2 [f] [p 2 f] 0. Thus p 1 f and p 2 f are homotopic to constant maps, say via homotopies h 1 t : I X and h 2 t : I Y, that is, h 1 0 p 1 f h 2 0 p 2 f h 1 1 c 1 h 2 1 c 2 for some constants c 1 X, c 2 Y. Then f is homotopic to a constant map via (s, t) (h 1 t (s), h 2 t (s)), since (s, 0) (h 1 0(s), h 2 0(s)) (p 1 f(s), p 2 f(s)) f(s) (s, 1) (h 1 1(s), h 2 1(s)) (c 1, c 2 ) Thus [f] 0, so ker φ is trivial, so φ is injective. This completes the proof that φ is an isomorphism. Lemma 0.3 (for topological group problem). Let X be a topological group with identity e. For loops f, g : I X based at e, define f g : I X by (f g)(s) f(s)g(s). This induces : π 1 (X, e) π 1 (X, e) π 1 (X, e) given by [f] [g] [f g]. We claim that this is well-defined. Furthermore, (f g) (f g ) (f f ) (g g ) Using for the usual multiplication in π 1 (X), we have ( [f] [f ] ) ( [g] [g ] ) ( [f] [g] ) ([f ] [g ] ) Proof. Suppose [f] [f ] and [g] [g], so we have homotopies f t : I X and g t : I X satisfying f 0 f f 1 f f t (0) f t (1) e g 0 g g 1 g g t (0) g t (1) e Then define h t : I X by h t (s) f t (s)g t (s). Then h 0 (s) f(s)g(s) (f g)(s) h 1 (s) f (s)g (s) (f g )(s) h t (0) f t (0)g t (0) e h t (1) f t (1)g t (1) e 3

4 Thus h t is a homotopy from f g to f g, so [f g] [f g ]. Thus the operation is well-defined. Now we compute ( (f g) (f g ) ) (f g)(2s) 0 s 1/2 (s) (f g )(2s 1) 1/2 s 1 f(2s)g(2s) 0 s 1/2 f (2s 1)g (2s 1) 1/2 s 1 ( ) ( ) f(2s) 0 s 1/2 g(2s) 0 s 1/2 f (2s 1) 1/2 s 1 g (2s 1) 1/2 s 1 ( f f (s) )( g g (s) ) Thus By reflexivity, from this we get ( (f f ) (g g ) ) (s) (f g) (f g ) (f f ) (g g ) (f g) (f g ) (f f ) (g g ) Thus ( [f] [f ] ) ( [g] [g ] ) ( [f] [g] ) ([f ] [g ] ) Lemma 0.4 (Eckmann-Hilton Theorem). Let X be a set with two binary operations,. Suppose that both operations have a unit, that is, there exist e, e X so that e x x x e e x x x e for all x X. Suppose also that for all w, x, y, z X we have (w x) (y z) (w y) (x z) Then, are equal, associative, and commutative. That is, for all x, y, z X, Proof. First, we show that e e. x y x y x y y x (x y) z x (y z) e e e (e e) (e e ) (e e) (e e ) e e e 4

5 Let x, y X. Then Thus the operations coincide. Also, x y (x e) (e y) (x e) (e y) x y x y (e x) (y e) (e y) (x e) y x Thus x y y x so the operations are commutative. Finally, Thus they are associative. x (y z) (x 1) (y z) (x y) (1 z) (x y) z Proposition 0.5 (written exercise from Prof. Hedden). Let X be a topological group. Then π 1 (X, e) is abelian. Proof. Define multiplication of paths elementwise as in Lemma 0.3 above. As shown in that lemma, ( [f] [f ] ) ( [g] [g ] ) ( [f] [g] ) ([f ] [g ] ) Thus, are binary operations on π 1 (X) satisfying the hypotheses of the Eckmann-Hilton Theorem, so they are equal and abelian. Thus the usual multiplication on π 1 (X) is abelian. Proposition 0.6 (Exercise , part one). Let A be a path-connected space. Form X by attaching an n-cell e n with n 2. Then the inclusion ι : A X induces a surjection on π 1. That is, ι : π 1 (A) π 1 (X) is surjective. Proof. Let f : S n 1 A be the attaching map. Then X A e n, where A and e n are path connected and open in X, and A e n f(s n 1 ) is also path-connected. Let x 0 f(s n 1 ). By Lemma 1.15 (Hatcher), every loop in X based at x 0 is homotopic to a product of loops, where each loop is either contained in e n or A. Since n 2, a loop contained in e n is nullhomotopic, so every loop in X is homotopic to a loop in A. Thus if [f] π 1 (X, x 0 ), there there is a loop f : I A so that [f ] [f]. We have f ι f, so Hence ι is surjective. ι [f ] [ι f ] [f ] [f] Proposition 0.7 (Exercise a). The wedge sum S 1 S 2 has fundamental group Z. Proof. As noted in Example 0.11 of Hatcher, S 1 S 2 can be formed by attaching S 2 to S 1 via a constant map. By the above, the inclusion ι : S 1 S 1 S 2 induces a surjection ι : π 1 (S 1 ) π 1 (S 1 S 2 ). By the first isomorphism theorem of groups, π 1 (S 1 S 2 ) π 1 (S 1 )/ ker ι Thus π 1 (S 1 S 2 ) is isomorphic to a quotient group of Z, so it is cyclic. Note that π 1 (S 1 S 2 ) is not finite, since it contains infinitely many non-homotopic loops (take loops winding n times around the S 1 part for n N). Thus π 1 (S 1 S 2 ) is infinite cyclic, that is, isomorphic to Z. 5

6 Proposition 0.8 (Exercise b). Let X be a path-connected CW complex with X 1 its 1-skeleton. Then the inclusion map ι : X 1 X induces a surjection ι : π 1 (X 1 ) π 1 (X). Proof. The space X is formed from X 1 by attaching n-cells for n 2. First, suppose there are finitely many cells e 1,..., e k. Let X 0 X 1, and define X i to be the CW complex formed after attaching e i to X i+1, so X k X. Then each inclusion ι i : X i X i+1 induces a surjection ι i : π 1 (X i ) π 1 (X i+1 ), so the (finite) composition ι k ι (k 1)... ι 1 ι 0 (ι k ι k 1... ι 1 ι 0 ) ι is surjective. Now suppose that X has infinitely many cells. Let [f] π 1 (X) and choose a representative loop f. The image of f is a compact subset of X, so by Proposition A.1 in the Appendix (Hatcher), the image is contained in a finite subcomplex X X. Let X 1 be the 1-skeleton of X, so X 1 X 1. Let ι : X 1 X be the inclusion. By the result in the finite case, ι : π 1 (X 1 ) π 1 (X) is surjective. Since [ the image of f lies in X, we can think of [f] π 1 (X). By surjectivity of ι, there exists f] π 1 (X 1 ) so [ ] that ι f [f]. Choose a representative f. Since the image of f is contained in X 1 X 1, [ we also have f] π 1 (X 1 ). Note that ι f ι f since the image of f lies in X 1. Thus Thus ι is surjective. [ ] [ ι f ι f ] [ ι f ] [ ] ι f [f] Proposition 0.9 (Exercise 1.3.1). Let p : X X be a covering map and A X, and let à p 1 (A). Then the restriction p à : à A is a covering map. Proof. Let a A. Since p is a covering map, there is an evenly covered neighborhood U so that a U X. That is, p 1 (U) α U α where each U α is mapped homeomorphically to U by p. By definition of subspace topology, U A is an open neighborhood of a in A. Using basic properties of preimages, ( ) p 1 (U A) p 1 (U) p 1 (A) U α à (U α Ã) α α Then p(u α Ã) p(u α) p(ã) p(u α) A U A so p : U Uα à α à U A is well-defined. First we claim that it is surjective. Let x U A. Since x U, there exists x α U α so that p(x α ) x., Since x A, x α Ã, 6

7 so x α U α Ã. Thus p U α à is surjective. It is injective since it is a restriction of the injective map p Uα : U α U. Thus it is bijective. It is continuous because it is a restriction of the continuous map p. It has a continuous inverse since its inverse is a restriction of the continuous inverse of p Uα. Thus p maps U Uα à α à homeomorphically onto U A. Since U A is a disjoint union of such sets, U A is an evenly covered neighborhood of a A. Hence p à is a covering map. Lemma 0.10 (for Exercise 1.3.2). Let p 1 : X1 X 1 and p 2 : X2 X 2 be set maps. Let U 1 X 1 and U 2 X 2. Then (p 1 p 2 ) 1 (U 1 U 2 ) p 1 1 (U 1 ) p 1 2 (U 2 ) Proof. This is a straightforward use of definitions. (p 1 p 2 )(U 1 U 2 ) ( x 1, x 2 ) X 1 X 2 : (p 1 p 2 )( x 1, x 2 ) U 1 U 2 } ( x 1, x 2 ) X 1 X 2 : (p 1 ( x 1 ), p 2 ( x 2 )) U 1 U 2 } ( x 1, x 2 ) X 1 X 2 : x 1 p 1 1 (U 1 ), x 2 p 1 2 (U 2 )} p 1 1 (U 1 ) p 1 2 (U 2 ) Proposition 0.11 (Exercise 1.3.2). Let p 1 : X 1 X 1 and p 2 : X 2 X 2 be covering maps. Then the product p 1 p 2 : X 1 X 2 X 1 X 2 is a covering map. Proof. Let (x 1, x 2 ) X 1 X 2. Since p 1, p 2 are covering maps, there exist evenly covered neighborhoods U 1 of x 1 and U 2 of x 2. That is, p 1 1 (U 1 ) U1 α α A p 1 2 (U 2 ) β B U β 2 where p 1 maps each U1 α homeomorphically to U 1 and p 2 maps each U β 2 homeomorphically to U 2. By definition of the product topology, U 1 U 2 is an open neighborhood of (x 1, x 2 ). Using the previous lemma, ( ) ( ) (p 1 p 2 ) 1 (U 1 U 2 ) p 1 1 (U 1 ) p 1 2 (U 2 ) U1 α U β 2 (U1 α U β 2 ) α A β B (α,β) A B Again, by definition of the product topology, U1 α U β 2 is open in X 1 X 2. We claim that (p 1 p 2 ) U α 1 U : U β 1 α U β 2 U 1 U 2 is a homeomorphism. 2 First we show that it is surjective. Let (x 1, x 2 ) U 1 U 2. Then by the even covering properties of p 1, p 2, there exist x 1 U1 α and x 2 U β 2 so that p 1 ( x 1 ) x 1 and p 2 ( x 2 ) x 2. 7

8 Thus ( x 1, x 2 ) U1 α U β 2 and (p 1 p 2 )( x 1, x 2 ) (x 1, x 2 ). This establishes surjectivity. Now suppose that there exist ( x 1, x 2 ), (ỹ 1, ỹ 2 ) U1 α U β 2 so that (p 1 p 2 )( x 1, x 2 ) (p 1 p 2 )(ỹ 1, ỹ 2 ) Then p 1 ( x 1 ) p 1 (ỹ 1 ) with x 1, ỹ 1 U1 α. But then by injectivity of p 1 on this domain, this implies x 1 ỹ 1. Similarly, x 2 ỹ 2. Hence p 1 p 2 is injective on this domain. Continuity of p 1 p 2 comes from the properties of the product topology, and continuity of the inverse on the restricted domain also comes from the properties of the product topology, this time on U1 α U β 2. Thus (p 1 p 2 ) U α 1 U : U β 1 α U β 2 U 1 U 2 is a homeomorphism. Thus U 1 U 2 is evenly 2 covered, so p 1 p 2 is a covering map. Proposition 0.12 (written exercise from Prof. Hedden). Not every local homeomorphism is a covering map. Proof. Let X x 1, x 2 } with the trivial topology, that is, the only open sets are, X. Let Y y 1, y 2, y 3 } with open sets, Y, y 1, y 2 }. (Note that this is a topology on Y.) Define f : X Y by f(x 1 ) y 1 and f(x 2 ) y 2. We claim that f is a local homeomorphism but not a covering map. First, note that f is continuous, since the preimages f 1 f 1 (Y ) X f 1 (y 1, y 2 }) X are open. It is a local homeomorphism because for either x 1, x 2 X, neighborhood U X gives the restriction f : x 1, x 2 } y 1, y 2 }, which is a homeomorphism. However, f is not a covering map. Consider the point y 3 Y. The only open neighborhood is Y itself, but the preimage of Y is X, which is a disjoint union of spaces mapped homeomorphically to X. 8