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Set Theory and Elementary Algebraic Topology

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Set Theory and Elementary Algebraic Topology SET THEORY AND ELEMENTARY ALGEBRAIC TOPOLOGY BY DR. ATASI DEB RAY DEPARTMENT OF MATHEMATICS WEST BENGAL STATE UNIVERSITY BERUNANPUKURIA, MALIKAPUR 24 PARAGANAS (NORTH) KOLKATA - 700126 WEST BENGAL, INDIA E-mail : [email protected] Chapter 6 Fundamental Group and its basic properties Module 1 Construction of Fundamental group and Induced homomorphism 1 2 6.1 Construction of Fundamental group and Induced homomorphism We begin with defining a binary operation on the collection Π1(X; x0). Theorem 6.1.1. Let (X; x0) be a pointed topological space and Π1(X; x0), the collection of all equivalence classes of loops in X based at x0, arising from the equivalence relation α ∼ β iff α is path homotopic to β. Then ([α]; [β]) 7! [α] ◦ [β] = [α ∗ β] is a well defined binary operation on Π1(X; x0). Proof. For any α; β 2 P, α ∗ β 2 P so that [α ∗ β] 2 Π1(X; x0). Let [α] = [α1] and [β] = [β1], where α, β, α1 and β1 2 P. Then α ∼ α1 and β ∼ β1. Now, α ∗ β ∼ α1 ∗ β1. Hence, [α] ◦ [β] = [α1] ◦ [β1], which shows that the operation is well defined. Theorem 6.1.2. For any pointed topological space (X; x0), (Π1(X; x0); ◦) is a group; where [α] ◦ [β] = [α ∗ β]. Proof. Follows directly from Theorems on path homotopy. Definition 6.1.1. The group Π1(X; x0), obtained in Theorem 6.1.2 is called the first fun- damental group or Poincar´e group of X based at x0. The definition of fundamental group depends on the base point x0. So, fundamental groups are associated to pointed topological spaces rather than topological spaces, in gen- eral. Later, we may convince ourselves by constructing examples of fundamental groups of topological spaces that depend on the base point. However, we shall see next that if x0 and x1 are two points lying in the same path component then their fundamental groups are isomorphic. Theorem 6.1.3. Let X be a topological space and x0; x1 2 X. If there is a path connecting x0 and x1 then Π1(X; x0) is isomorphic to Π1(X; x1). Proof. Let ! : [0; 1] ! X be a path connecting x0 and x1. Then !(0) = x0 and !(1) = x1. −1 Define f! :Π1(X; x0) ! Π1(X; x1) by f!([α]) = [! ∗ α ∗ !]. We first see that this map is −1 −1 well defined. Let [α] = [β]. Then α ∼x0 β and therefore, ! ∗ α ∗ ! ∼x1 ! ∗ β ∗ !. So, f!([α]) = f!([β]). If α ∼x0 α1 and β ∼x0 β1 then α ∗ α1 ∼x0 β ∗ β1. Using this and a Theorem that we have obtained in Chapter 5, we get f!([α] ◦ [β]) = f!([α ∗ β]) = [!−1 ∗ (α ∗ β) ∗ !] −1 = [! ∗ (α ∗ Cx0 ∗ β) ∗ !] = [!−1 ∗ (α ∗ ! ∗ !−1 ∗ β) ∗ !] = [!−1 ∗ α ∗ !] ◦ [!−1 ∗ β ∗ !] = f!([α]) ◦ f!([β]): 3 Hence, f! is a homomorphism. If f!−1 :Π1(X; x1) ! Π1(X; x0) is defined by f!−1 ([γ]) = −1 [! ∗ γ ∗ ! ], then it is also a group homomorphism. We see that f!−1 and f! are inverse of each other : −1 −1 −1 (f! ◦ f! )([α]) = f! ([! ∗ α ∗ !]) = [! ∗ (!−1 ∗ α ∗ !) ∗ !−1] = [Cx0 ∗ α ∗ Cx0 ] = [α] ∼ for all [α] 2 Π1(X; x0). This establishes, f! is an isomorphism. Consequently, Π1(X; x0) = Π1(X; x1). ∼ Corollary 6.1.1. If X is path connected then for any two points x0, x1 2 X,Π1(X; x0) = Π1(X; x1). Henceforth, in view of the last result, we denote the fundamental group of a path con- nected space X by Π1(X), without mentioning the base point explicitly. Theorem 6.1.4. If X is a topological space and !, η are homotopic paths joining x0, x1 2 X, then f! = fη. −1 −1 −1 −1 Proof. Since ! ∼ η ) ! ∼ η , we get ! ∗ α ∗ ! ∼ η ∗ α ∗ η , for any [α] 2 Π1(X; x0). −1 −1 Hence, fη([α]) = [η ∗ α ∗ η ] = [! ∗ α ∗ ! ] = f!([α]). The following result gives a necessary and sufficient condition for Π1(X; x0) to be abelian. Theorem 6.1.5. Let X be a path connected space and x0; x1 2 X.Π1(X; x0) is abelian if and only if for each pair of paths ! and η from x0 to x1, f! = fη. −1 −1 Proof. Suppose Π1(X; x0) is abelian. Since !∗η is a loop based at x0,[!∗η ] 2 Π1(X; x0). −1 −1 −1 −1 Hence, [! ∗ η ] ◦ [α] = [α] ◦ [! ∗ η ], for all [α] 2 Π1(X; x0). Then ! ∗ η ∗ α ∼ α ∗ ! ∗ η which implies that η−1 ∗ α ∗ η ∼ !−1 ∗ α ∗ !. i.e., [η−1 ∗ α ∗ η] = [!−1 ∗ α ∗ !] and so, fη([α]) = f!([α]) for all [α] 2 Π1(X; x0). Conversely, let the condition hold and [α]; [β] 2 Π1(X; x0). Since X is path connected, there is a path ! (say) from x0 to x1. Then β ∗ ! is also a path from x0 to x1. By the −1 hypothesis, f! = fβ∗!. i.e., for all [α] 2 Π1(X; x0), f!([α]) = fβ∗!([α]). So, [! ∗ α ∗ !] = [(β ∗ !)−1 ∗ α ∗ (β ∗ !)] = [(!−1 ∗ β−1) ∗ α ∗ (β ∗ !)]. So, !−1 ∗ α ∗ ! ∼ !−1 ∗ (β−1 ∗ α ∗ β) ∗ ! which gives α ∼ β−1 ∗ α ∗ β, i.e., β ∗ α ∼ α ∗ β. Hence, [β ∗ α] = [α ∗ β]. Definition 6.1.2. A topological space X is called simply connected if it is path connected and its fundamental group is trivial. In what follows, we shall mean a trivial fundamental group when we write Π1(X; x0) = 0. We observe next that a continuous function between pointed topological spaces induces a homomorphism between the corresponding fundamental groups. Theorem 6.1.6. If f :(X; x0) ! (Y; y0) is a continuous function between the pointed topological spaces (X; x0) and (Y; y0), then there exists a homomorphism f# :Π1(X; x0) ! Π1(Y; y0). 4 Proof. By definition of continuous function between pointed spaces, we have f(x0) = y0. Define f# :Π1(X; x0) ! Π1(Y; f(x0)) by f#([α]) = [f ◦ α]. We show that such f# is well defined. Suppose [α] = [β] 2 Π1(X; x0). Then α ∼x0 β. Consequently, f ◦ α ∼f(x0) f ◦ β. i.e., f#([α]) = [f ◦ α] = [f ◦ β] = f#([β]). For any [α]; [β] 2 Π1(X; x0) and t 2 [0; 1], f ◦ (α ∗ β)(t) = f((α ∗ β)(t)) ( 1 f(α(2t)); 0 ≤ t ≤ 2 = 1 f(β(2t − 1)); 2 ≤ t ≤ 1 = ((f ◦ α) ∗ (f ◦ β))(t) Hence, [f ◦ (α ∗ β)] = [(f ◦ α) ∗ (f ◦ β)]. So, f#([α] ◦ [β]) = f#([α ∗ β]) = [f ◦ (α ∗ β)] = [(f ◦ α) ∗ (f ◦ β)] = [f ◦ α] ◦ [f ◦ β] = f#([α]) ◦ f#([β]): Therefore, f# is a homomorphism. The function f#, defined in Theorem 6.1.6 is called the homomorphism induced by f. The following are two important properties of the induced homomorphism. Theorem 6.1.7. (i) If I :(X; x0) ! (X; x0) is the identity map then I# :Π1(X; x0) ! Π1(X; x0) is the identity homomorphism. (ii) If f :(X; x0) ! (Y; y0) and g :(Y; y0) ! (Z; z0) are continuous maps then (g ◦ f)# = g# ◦ f#. Proof. (i) For each [α] 2 Π1(X; x0), (I ◦ α)(t) = α(t), for all t 2 [0; 1] and hence, we have I#([α]) = [I ◦ α] = [α]. So, the induced homomorphism I# :Π1(X; x0) ! Π1(X; x0) is the identity map. (ii) For all [α] 2 Π1(X; x0), (g ◦ f)#([α]) = [(g ◦ f) ◦ α] = [g ◦ (f ◦ α)] = g#([f ◦ α]) = g#(f#([α])) = (g# ◦ f#)([α]) Hence, (g ◦ f)# = g# ◦ f#. In view of Theorem 6.1.7, we conclude that Π1 is a covariant functor Π1 : T op∗ ) Grp described as follows: (X; x0) 7−! Π1(X; x0) f f# (Y; y0) 7−! Π1(Y; y0) The properties mentioned in Theorem 6.1.7 are referred to as functorial properties of Π1. 5 Exercise 6.1.1. Let X be a topological space and a 2 A ⊆ X. Prove that (i) the induced homomorphism r# :Π1(X; a) ! Π1(A; a) is surjective if r : X ! A is a retraction; (ii) the induced homomorphism i# :Π1(A; a) ! Π1(X; a) is injective if i : A ! X is the inclusion map. Solution. (i) By definition of induced homomorphism r#([α]) = [r ◦ α]. Let [β] 2 Π1(A; a). Then [i ◦ β] 2 Π1(X; a), where i : A ! X is the inclusion map. Clearly, r#([i ◦ β]) = [β], proving the map to be surjective. (ii) Follows immediately from injectiveness of the inclusion map..
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