The Fundamental Group JWR Feb 6, 2005 1 The Fundamental Group 1.1. When γ : I ! X is a path, we denote by [γ] the homotopy class of γ (rel @I); here @I = f0; 1g. Two paths ®; ¯ : I ! X with ®(1) = ¯(0) determine a path ® ¢ ¯ : I ! X via the formula ( 1 ®(2t); 0 · t · 2 ; (® ¢ ¯)(t) := 1 ¯(2t ¡ 1); 2 · t · 1: The fundamental group of the pointed space (X; x0) is ¼1(X; x0) = f[γ]: γ(0) = γ(1) = x0g: The group operation is well defined by [®][¯] := [® ¢ ¯]. The identity element is the constant path and the inverse operation is defined by [γ]¡1 := [¯γ]; γ¯(t) := γ(1 ¡ t): Note that any path ® : I ! X gives an isomorphism ¼1(X; x0) ! ¼1(X; x1):[γ] 7! [¯® ¢ γ ¢ ®] where x0 = ®(0), x1 = ®(1), and (as before)γ ¯(t) := γ(1 ¡ t). In this sense, if X is path connected, the fundamental group is independent of the choice of the base point. Remark 1.2. A groupoid is an algebraic structure consisting of two sets B (the objects) and G (the morphisms) and five maps s; t : G ! B (source and target), e : B ! G (identity), m : G s£t G ! G (composition), and i : G ! G (inverse) satisfying the following axioms: (Identity Law) g ¢ ex = g, ey ¢ g = g where s(g) = x, t(g) = y. (Associative Law) g1 ¢ (g1 ¢ g3) = (g1 ¢ g1) ¢ g3 for g1; g2; g3 2 G s£t G. ¡1 ¡1 (Inverse Law) g ¢ g = gey, g ¢ g = ex where s(g) = x, t(g) = y. 1 Here we are using the abbreviations G s£t G := f(g1; g2) 2 G £ G : t(g2) = s(g1)g; ¡1 ex := e(x); g1 ¢ g2 := m(g1; g2); g := i(g): If the map i and the Inverse Law are dropped we recover the definition of a category. In a category a morphism which has a (necessarily unique) two-sided inverse is called an isomorphism (or an automorphism if source and target are the same). Thus a groupoid is a category in which every morphism is an isomorphism. For a category (B; G) and x; y 2 B denote the set of morphisms from x to y by Gx;y := fg 2 G : s(g) = x; t(g) = yg: The set Gx := fg 2 Gx;x : g is an isomorphismg is called the automorphism group of the object x. With these definitions ¼1(X; x0) is the automorphism group x0 in the fundamental groupoid defined by B = X; G = f [γ] j γ : I ! Xg; s([γ]) = γ(0), t([γ]) = γ(1), e(x) = [x] (the homotopy class of the constant path at the point x), m([¯]; [®]) = [® ¢ ¯], and i(γ) =γ ¯. Definition 1.3. A path connected space is called simply connected iff its fundamental group ¼1(X; x0) is trivial. 1 n Theorem 1.4. ¼1(S ) = Z and S is simply connected for n > 1. 2 Covering Spaces 2.1. A map p : E ! B is called locally trivial iff every x0 2 B there is a ¡1 neighborhood U of x0 2 B and a homeomorphism Á : U £ F ! p (U) such that p(Á(x; v)) = x for (x; v) 2 U £ F . The map p is called the projection, E is called the total space, B is called the base, and the topological space F is called the fiber. It is easy to see that if B is connected, then the fiber is unique up to homeomorphism. A locally trivial map is also called a fiber bundle, but sometimes the latter term has a more restricted meaning. A covering space is a locally trivial map with discrete fiber. Usually we denote covering spaces with notation like p : Y ! X or p : X˜ ! X. Note that the projection p of a covering space is a local homeomorphism. Example 2.2. The archetypal example of a covering space is the exponential map p : R ! S1 defined by p(θ) = eiθ. The restriction of p to an open interval is a local homeomorphism but not a covering. Example 2.3. The map p : S1 ! S1, p(z) = zn is an n-sheeted covering space. Example 2.4. The various covers of a wedge S1 _S1 of circles shown on page 58 of [3] are a good source of examples. 2 2.5. When p : X˜ ! X and f : Y ! X we call a map f˜ : Y ! X˜ a lift of f iff ˜ p ± f = f. If p :(X;˜ x˜0) ! (X; x0) and f :(Y; y0) ! (X; x0) a pointed lift of ˜ ˜ f is a lift f such that f(y0) =x ˜0. Proposition 2.6 (Path Lifting). Let p :(X;˜ x˜0) ! (X; x0) is a covering ¡1 space, γ : I ! X, and y0 2 p (γ(0)). Then there is a unique lift γ˜ : I ! X˜ of γ with γ˜(0) = y0. Corollary 2.7 (Homotopy Lifting). Let p : X˜ ! X is a covering space, fft : Y ! Xgt2I be a homotopy, and g : Y ! X˜ of be a lift of f0. Then there ˜ ˜ ˜ is a unique homotopy fft : I ! X˜gt2I such that f0 = g and ft is a lift of ft. ˜ Proof. The uniqueness of ft(y) follows from 2.6 applied to the path I 7! X : ˜ t 7! ft(y) with starting point g(y) 2 X˜ and this also defines ft(y) for y 2 Y and ˜ t 2 I. We show that (t; y) 7! ft(y) is continuous at (t0; y0). Bu compactness write write the interval [0; t0] as a union of closed intervals [0; t0] = I1 [ ¢ ¢ ¢ [ Ik so that ft(y0) 2 Ui for t 2 Ii where Ui ½ X is such that p is trivial over Ui. By induction on i there are open sets U˜i ½ X˜ such that p maps U˜i homeomorphically ˜ onto Ui and ft(y0) 2 U˜i for t 2 Ii. By continuity there is a neighborhood V of y0 2 Y such that ft(y) 2 Ui for t 2 Ii and y 2 V . By induction on i, the fact that p : U˜i ! Ui is a homeomorphism, and uniqueness of path lifting, ˜ ft(y) 2 U˜i for t 2 Ii and y 2 V . Now, since p : U˜i ! Ui is a homeomorphism, ˜ the continuity of [0; t0] £ V ! X˜ :(t; y) 7! ft(y) at points t 2 [0; t0] and y 2 V follows from the continuity of [0; t0] £ V ! X :(t; y) 7! ft(y) Corollary 2.8 (Injectivity). Let p :(X;˜ x˜0) ! (X; x0) is a covering space. Then the induced map p¤ : ¼1(X;˜ x˜0) ! ¼1(X; x0) is injective. Theorem 2.9 (Lifting Criterion). Assume p :(X;˜ x˜0) ! (X; x0) be a cover- ing space and let f :(Y; y0) ! (X; x0) be a map. Assume Y is path connected and locally path connected. Then (i) Two pointed lifts of the same map f are equal. (ii) There is a pointed lift of f if and only if f¤¼1(Y; y0) ½ p¤¼1(X;˜ x˜0). Remark 2.10. Since p¤ is injective, we have f¤¼1(Y; y0) ½ p¤¼1(X;˜ x˜0) if and only if there is a homomorphism h : ¼1(Y; y0) ! ¼1(X;˜ x˜0) such that p¤ ±h = f¤. ˜ When these equivalent conditions hold we have h = f¤ as follows: If p¤ ± h = f¤, ˜ ˜ then p¤ ± h = f¤ = p¤ ± f¤ so h = f¤ by the injectivity of p¤. Thus, in the two diagrams (X;˜ x˜0) ¼1(X;˜ x˜0) s9 p8 f˜ ss h pp ss p pp p ss ppp ¤ sss ² ppp ² f f¤ (Y; y0) / (X; x0) ¼1(Y; y0) / ¼1(X; x0) we can find f˜ so that the diagram on the left commutes if and only if we can find h so that the diagram on the right commutes. 3 ˜ Proof of Theorem 2.9. When f is a pointed lift of f and ® is a path from y0 to ˜ ˜ y, then f ± ® is a path from f(y0) to f(y) and f ± ® is a path from f(y0) to f˜(y), so uniqueness of the lift follows from uniqueness of path lifting. ’Only if’ ˜ ˜ in part (ii) follows from (p ± f)¤ = p¤ ± f¤. We prove ’if’. ˜ Assume that f¤¼1(Y; y0) ½ p¤¼1(X;˜ x˜0). As just noted, we must define f(y) ˜ by f(y) =γ ˜(1) whereγ ˜ is the lift of γ := f ± ® withγ ˜(0) =x ˜0 and ® is a path ˜ from y0 to y in Y . We must show that f is well defined (i.e. independent of the choice of ®) and continuous. ¯ Suppose that ¯ is another path from y0 to y. Then [®¯] 2 ¼1(Y; y0) so ¯ [(f ± ®)(f ± ¯)] 2 f¤¼1(Y; y0) ½ p¤¼1(X;˜ x˜0) so there is an element of [˜γ] 2 ¯ ¯ ¼1(X;˜ x˜0) which projects to [(f ±®)(f ±¯)], i.e. p±γ˜ is homotopic to (f ±®)(f ±¯) with endpoints fixed so the liftγ ˜ of which starts atx ˜0 has the same endpoint ¯ as the lift of (f ± ®)(f ± ¯) of which starts atx ˜0. But this endpoint isx ˜0 asγ ˜ is ¯ closed. Hence the lift of (f ± ®)(f ± ¯) of which starts atx ˜0 is closed so the lifts of f ± ® and f ± ¯ which start atx ˜0 have the same endpoint as required. ˜ Now we show that f is continuous. Choose y1 2 Y and a neighborhood W˜ ˜ ˜ of f(y1) in X˜. We must construct a neighborhood V of y1 such that f(V ) ½ W˜ .