FUNDAMENTAL GROUPS AND THE VAN KAMPEN'S THEOREM ANG LI Abstract. In this paper, we start with the definitions and properties of the fundamental group of a topological space, and then proceed to prove Van- Kampen's Theorem, which helps to calculate the fundamental groups of com- plicated topological spaces from the fundamental groups we know already. We also use the properties of covering space to prove the Fundamental Theorem of Algebra and Brouwer's Fixed Point Theorem. Contents 1. Homotopies and the Fundamental Group 1 2. Deformation Retractions and Homotopy type 6 3. Van Kampen's Theorem 9 4. Applications of van Kampen's Theorem 13 5. Fundamental Theorem of Algebra 14 6. Brouwer Fixed Point Theorem 15 Acknowledgments 16 References 16 1. Homotopies and the Fundamental Group One of the fundamental problems in topology is to determine whether two topo- logical spaces are homeomorphic. Though there is no general method to solve this, but there are techniques to determine this for special cases. Showing that two topological spaces are homeomorphic is always done by con- structing a homeomorphism between them and we have techniques help us to con- struct homeomorphisms. However, showing that two spaces are not homeomorphic is more tricky because it is impossible to show that any continuous function is not a homeomorphism be- tween these two spaces. Thus, to show that two spaces are not homeomorphic, we look at properties of topological spaces that are preserved under homeomorphism. One of these invariants is the fundamental group of a topological space. If two topological spaces have different fundamental groups, then they are not homeo- morphic because as we will see, two homeomorphic spaces should have isomorphic fundamental groups. Date: September 9, 2013. 1 2 ANG LI Before defining the fundamental group of a space X, we shall define an equiva- lence relation called path homotopy. Definition 1.1. If f and f 0 are continuous maps of the space X into the space Y , we say that f is homotopic to f 0 if there is a continuous map F : X × I ! Y such that (1.2) F (x; 0) = f(x) and F (x; 1) = f 0(x) for each x 2 X. The map F is called a homotopy between f and f 0. If f is homotopic to f 0, we write f ' f 0. If f ' f 0 and f 0 is a constant map, we say that f is nulhomotopic. We can think of the parameter t as a representation of time, a homotopy between f and f 0 represents a continuous transformation from f to f 0. There is a special case when f is a path in X, connecting x0 and x1, i.e. (1.3) f : [0; 1] ! X is a continuous map such that f(0) = x0 and f(1) = x1. x0 is called the initial point and x1 is called the final point. Definition 1.4. Two paths f and f 0 are said to be path homotopic if they have the same initial point x0 and final point x1, and if there is a continuous map F : I × I ! X such that (1.5) F (s; 0) = f(s) and F (s; 1) = f 0(s); (1.6) F (0; t) = x0 and F (1; t) = x1; for each s 2 I and each t 2 I. We call F a path homotopy between f and f 0. If 0 0 f is path homotopic to f , we write f 'p f . The first condition is simply that F is a homotopy between f and f 0, and the second says that for fixed t, F (s; t) is a path from x0 to x1. Example 1.7. One example of homotopy is the straight-line homotopy. Let f and g be two continuous maps of a space X into R2. Then f and g are homotopic with the following homotopy: (1.8) F (x; t) = (1 − t)f(x) + tg(x): This homotopy between f and g is called the straight-line homotopy for the fact that it moves from f(x) to g(x) along the straight line connecting them. Example 1.9. However, in the previous example, if we replace R2 by the punctured plane, R2−0, which is not convex, the straight-line homotopy wouldn't always work. Consider the following paths in punctured plane: (1.10) f(s) = (cosπs; sinπs) and g(s) = (cosπs; 2sinπs): f and g are path homotopic and the straight-line homotopy between them is ac- ceptable. But the straight-line homotopy between f and the path (1.11) h(s) = (cosπs; −sinπs) is not acceptable because the straight-line would pass the origin, which is not in the punctured plane. FUNDAMENTAL GROUPS AND THE VAN KAMPEN'S THEOREM 3 Figure 1. Straight-line Homotopy Indeed, there no path homotopy between f and h but it takes some work to prove. Figure 2. Punctured Plane Directly from the definitions of homotopy and path homotopy, we have the fol- lowing lemma, whose proof is trivial. Lemma 1.12. The relations ' and 'p are equivalence relations. Because 'p is an equivalence relation, we can define a certain operation on path-homotopy classes as follows: Definition 1.13. If f is a path in X from x0 to x1, and if g is a path in X from x1 to x2, we define the product f ∗ g of f and g to be the path h given by the equations ( f(2s); for s 2 [0; 1=2]: (1.14) h(x) = g(2s − 1) for s 2 [1=2; 1]: The function h is well-defined and continuous, by the pasting lemma and it is a path in X from x0 to x2. 4 ANG LI This product operation on paths induces a well-defined operation on equivalence classes, defined by (1.15) [f] ∗ [g] = [f ∗ g] We can prove that the operation ∗ between equivalence classes is associative with left and right identities as well as inverse for any equivalence class [f]. A result that will be useful to us later is that [f] can be decomposed to the product of equivalence classes of the segments composing f. Theorem 1.16. Let f be a path in X, and let a0, ..., an be numbers such that 0 = a0 < a1 < ::: < an = 1. Let fi : I ! X be the path that equals the positive linear map of I onto [ai−1; ai] followed by f. Then (1.17) [f] = [f1] ∗ [f2] ∗ ::: ∗ [fn]: With this operator ∗ on equivalence classes of 'p, we just have a groupoid instead of a group because we couldn't apply this operator on any two equivalence classes for the fact that they don't necessarily satisfy the end point condition required to apply ∗. However, if we pick out the equivalence classes with the same initial and final point, with the operator ∗, we could construct a group. Definition 1.18. Let X be a space; let x0 be a point of X. A path in X that begins and ends at x0 is called a loop based at x0. The set of path homotopy classes of loops based at x0, with the operation ∗ is called the fundamental group of X relative to the base point x0. It is denoted by π1(X; x0). n Example 1.19. Let R be the euclidean n-space. Then π1(X; x0) is trivial for x0 2 X because if f is a loop based at x0, the straight-line homotopy is a path homotopy between f and the constant path at x0. More generally, if X is any n convex subset of R , then π1(X; x0) is the trivial group. A well-known result is that the fundamental group of the circle is isomorphic to Z, which can be proved by covering space theory. We state it as a theorem but omit the proof here. Theorem 1.20. The fundamental group of the circle is isomorphic to the additive group of the integers, i.e. 1 ∼ (1.21) π1(S ) = Z: One immediate question is that the fundamental group depends on the base point. We now consider this question. Definition 1.22. Let α be a path in X from x0 to x1. We define a map (1.23)α ^ : π1(X; x0) ! π1(X; x1) by (1.24)α ^([f]) = [¯α] ∗ [f] ∗ [α]; whereα ¯(t) = α(1 − t). If f is path from x0 to x1, then the image of [f] underα ^ is a loop based at x1. Henceα ^ maps π1(X; x0) to π1(X; x1) as desired. Theorem 1.25. The map α^ is a group isomorphism. FUNDAMENTAL GROUPS AND THE VAN KAMPEN'S THEOREM 5 Proof. To show thatα ^ is a homomorphism, we compute α^([f]) ∗ α^([g]) = ([¯α] ∗ [f] ∗ [α]) ∗ ([¯α] ∗ [g] ∗ [α])(1.26) = [¯α] ∗ [f] ∗ [g] ∗ [α](1.27) (1.28) =α ^([f] ∗ [g]): To show thatα ^ is an isomorphism, we show that α¯^ is an inverse forα ^. (1.29) β^([h]) = [α] ∗ [h] ∗ [¯α]; (1.30)α ^(β^([h])) = [¯α] ∗ ([α] ∗ [h] ∗ [¯α]) ∗ [α] = [h]: ^ Similarly, we have β(^α[f]) = [f] for each [f] 2 π1(X; x0). And thus,α ^ is a group isomorphism. Using this result, we can prove that if a space X is path connected, all the fundamental groups π1(X; x) are isomorphic because there is a path connecting any two points in X. Definition 1.31. A space X is said to be simply connected if it is a path- connected space and if π1(X; x0) is trivial group for some x0 2 X, and hence for every x0 2 X.