Chapter 8 The Fundamental Group and Covering Spaces In the first seven chapters we have dealt with point-set topology. This chapter pro- vides an introduction to algebraic topology. Algebraic topology may be regarded as the study of topological spaces and continuous functions by means of algebraic objects such as groups and homomorphisms. Typically, we start with a topological ᐀ space (X, ) and associate with it, in some specific manner, a group GX (see Appen- dix I). This is done in such a way that if (X, ᐀) and (Y, ᐁ) are homeomorphic, then → GX and GY are isomorphic. Also, a continuous function f : X Y “induces” a homo- → morphism f : GX GY, and, if f is a homeomorphism, then f* is an isomorphism. * ᐀ Then the “structure” of GX provides information about the “structure” of (X, ). One of the purposes of doing this is to find a topological property that distinguishes between two nonhomeomorphic spaces. The group that we study is called the funda- mental group. 8.1 Homotopy of Paths In Section 3.2 we defined paths and the path product of two paths. This section stud- ies an equivalence relation defined on a certain collection of paths in a topological space. Review Theorems 2.15 and 2.27, because these results about continuous func- tions will be used repeatedly in this chapter and the next. The closed unit interval [0, 1] will be denoted by I. Definition. Let (X, ᐀) and (Y, ᐁ) be topological spaces and let f, g: X → Y be con- tinuous functions. Then f is homotopic to g, denoted by f Ӎ g, if there is a continu- ous function H: X ϫ I → Y such that H(x, 0) ϭ f(x) and H(x, 1) ϭ g(x) for all x ෈ X. The function H is called a homotopy between f and g. ᭿ 265 266 Chapter 8 ■ The Fundamental Group and Covering Spaces In the preceding definition, we are assuming that I has the subspace topology ᐂ determined by the usual topology on ޒ . Thus X ϫ I has the product topology deter- mined by ᐀ and ᐂ . We think of a homotopy as a continuous one-parameter family of continuous functions from X into Y. If t denotes the parameter, then the homotopy represents a continuous “deformation” of the function f to the function g as t goes from 0 to 1. The question of whether f is homotopic to g is a question of whether there is a con- tinuous extension of a given function. We think of f as being a function from X ϫ {0} into Y and g as being a function from X ϫ {1} into Y, so we have a continuous func- tion from X ϫ {0, 1} into Y and we want to extend it to a continuous function from X ϫ I into Y. EXAMPLE 1. Let X and Y be the subspaces of ޒ ϫ ޒ defined by X ϭ {(x, y) ෈ ޒ ϫ ޒ: x2 ϩ y2 ϭ 1} and Y ϭ {(x, y) ෈ ޒ ϫ ޒ: (x ϩ 1)2 ϩ y2 ϭ 1 or (x Ϫ 1)2 ϩ y2 ϭ 1}. Define f, g: X → Y by f(x, y) ϭ (x Ϫ 1, y) and g(x, y) ϭ (x ϩ 1, y). (See Figure 8.1). g X Y f Figure 8.1 Then f is not homotopic to g. EXAMPLE 2. Let X be the subspace of ޒ ϫ ޒ defined by X ϭ {(x, y) ෈ ޒ ϫ ޒ: x2 ϩ y2 ϭ 1} and let Y ϭ X ϫ X. Define f, g: X → Y by f(x, y) ϭ ((1, 0), (x, y)) andg (x, y) ϭ ((0,1), (x, y)). Then the function H: X ϫ I → Y defined by H((x, y), t) ϭ ((͙1 Ϫ t2, t), (x, y)) is a homotopy between f and g so f Ӎ g. Notice that X is a circle and Y is a torus, f “wraps” the circle X around {(1, 0)}ϫ X, and g “wraps” the circle X around {(0, 1)} ϫ X (see Figure 8.2). Let A denote the arc from ((1, 0), (1, 0)) to ((0, 1), (1, 0)) (see Figure 8.2). Then H maps X ϫ I onto A ϫ X. 8.1 Homotopy of Paths 267 A (0, 1) g ((0, 1), (1, 0)) (1, 0) H | (( 3/2, 1/2), (1, 0)) (X 3 {1/2}) f ((1, 0), (1, 0)) Figure 8.2 THEOREM 8.1. Let (X, ᐀) and (Y, ᐁ) be topological spaces. ThenӍ is an equivalence relation on C(X, Y) (the collection of continuous functions that map X into Y). Proof. Let f ෈ C(X, Y). Define H: X ϫ I → Y by H(x, t) ϭ f(x) for each (x, t) ෈ X ϫ I. Then H is continuous and H(x, 0) ϭ H(x, 1) ϭ f(x), so f Ӎ f. Suppose f, g ෈ C(X, Y) and f Ӎ g. Then there is a continuous function H: X ϫ I → Y such that H(x, 0) ϭ f(x) and H(x, 1) ϭ g(x) for all x ෈ X. Define K: X ϫ I → Y by K(x, t) ϭ H(x, 1 Ϫ t). Then K is continuous, and K(x, 0) ϭ H(x, 1) ϭ g(x) and K(x, 1) ϭ H(x, 0) ϭ f(x) for all x ෈ X. Therefore g Ӎ f. Suppose f, g, h ෈ C(X, Y), f Ӎ g, and g Ӎ h. Then there are continuous func- tions F, G: X ϫ I → Y such that F(x, 0) ϭ f(x), F(x, 1) ϭ g(x), G(x, 0) ϭ g(x), and G(x, 1) ϭ h(x) for all x ෈ X. Define H: X ϫ I → Y by F(x, 2t), for all x ෈ X and 0 Յ t Յ 1 H(x, t) ϭ 2 { ෈ 1 G(x, 2t Ϫ 1), for all x X and 2 Յ t Յ 1. Then H is continuous, and H(x, 0) ϭ F(x, 0) ϭ f(x) and H(x, 1) ϭ G(x, 1) ϭh(x) for all x ෈ X. Therefore f Ӎ h. ᭿ If ␣ and ␤ are paths in a topological space, we define a relation between them that is stronger than homotopy. Definition. Let ␣ and ␤ be paths in a topological space (X, ᐀). Then ␣ is path ␤ ␣ Ӎ ␤ ␣ ␤ homotopic to , denoted by p , if and have the same initial point x0 → and the same terminal point x1 and there is a continuous function H: I ϫ I X ␣ ␤ ෈ such that H(x, 0) ϭ (x) and H(x, 1) ϭ (x) for all x X, and H(0, t) ϭ x0 ෈ andH(1, t) ϭ x1 for all t I. The function H is called a path homotopy between f and g. ᭿ 268 Chapter 8 ■ The Fundamental Group and Covering Spaces The question of the existence of a path homotopy between two paths is again a question of obtaining a continuous extension of a given function. This time we have a continuous function on (I ϫ {0, 1}) ʜ ({0, 1} ϫ I) and we want to extend it to a continuous function on I ϫ I (see Figure 8.3). X I 3 I x0 x1 Figure 8.3 EXAMPLE 3. Let X ϭ {(x, y) ෈ ޒ2: 1 Յ x2 ϩ y2 Յ 4} (see Figure 8.4), let ␣: I → X be the path that maps I in a “linear” fashion onto the arc A from (Ϫ1, 0) to (1, 0), and let ␤: I → X be the path that maps I in a “linear” fashion onto the arc B from (Ϫ1, 0) to (1, 0). Then ␣ is homotopic to ␤ (by a homotopy that transforms ␣ into ␤ in the manner illustrated in Figure 8.5), but ␣ is not path homotopic to ␤ because we cannot “get from” ␣ to ␤ and keep the “end points” fixed without “cross- ing the hole” in the space. ᐀ ෈ ␣ → ␣ If (X, ) is a topological space and x0 X, let ⍀(X, x0) ϭ { : I X: is con- ␣ ␣ ␣ ␤ ෈ tinuous and (0) ϭ (1) ϭ x0}. Observe that if , ⍀(X, x0), then the product ␣ ␤ ෈ (defined in Section 3.2) * ⍀(X, x0). It is, however, easy to see that * is not associative on ⍀(X, x0). Each member of ⍀(X, x0) is called a loop in X at x0. The proof of the following theorem is similar to the proof of Theorem 8.1 and hence it is left as Exercise 1. A (21, 0) (1, 0) B Figure 8.4 8.1 Homotopy of Paths 269 ␤(I) ␤(I) ␤(I) ␣(I) ␣ (I) ␣(I) ␤(I) ␤(I) ␤(I) ␣(I) ␣(I) ␣(I) Figure 8.5 ᐀ ෈ Ӎ THEOREM 8.2. Let (X, ) be a topological space, and let x0 X. Then p is an ᭿ equivalence relation on ⍀(X, x0). ᐀ ෈ ␣ ෈ ␣ Let (X, ) be a topological space and let x0 X. If ⍀(X, x0), we let [ ] ␣ ␲ denote the path-homotopy equivalence class that contains . Then we let 1(X, x0) denote the set of all path-homotopy equivalence classes on ⍀(X, x0), and we define ␤ ␣ ␤ ؠ ␲ ␣ ؠ an operation on 1(X, x0) by [ ] [ ] ϭ [ * ]. The following theorem tells us .thatؠ is well-defined ᐀ ෈ THEOREM 8.3. Let (X, ) be a topological space and let x0 X. Furthermore, ␣ ␣ ␤ ␤ ෈ ␣ Ӎ ␣ ␤ Ӎ ␤ ␣ ␤ Ӎ let 1, 2, 1, 2 ⍀(X, x0) and suppose 1 p 2 and 1 p 2. Then 1 * 1 p ␣ ␤ 2 * 2. ␣ Ӎ ␣ ␤ Ӎ ␤ Proof. Since 1 p 2 and 1 p 2, there exist continuous functions F, G: I ϫ → ␣ ␣ ␤ I X such that F(x, 0) ϭ 1(x), F(x, 1) ϭ 2(x), G(x, 0) ϭ 1(x), and G(x, 1) ϭ ␤ ෈ ෈ 2(x) for all x I, and F(0, t) ϭ F(1, t) ϭ G(0, t) ϭ G(1, t) ϭ x0 for all t I.