THE FUNDAMENTAL GROUP AND ITS APPLICATIONS PUBO HUANG April 2021 Advised by Professor Monty McGovern Senior Thesis submitted in partial fulfillment for the honors requirement for the Bachelor of Science degree in Mathematics. Introduction This is an exposition of the fundamental group, a powerful tool in algebraic topology|what it is, why it matters, and what it can do. We will mainly follow [Mun00b] and [Hat00a]. In chapter 1, we start by introducing some basic notions of homotopies, then formally define the fundamental group. In chapter 2, we introduce covering spaces, which will allow us to compute the fundamental groups of some simple spaces. In chapter 3, we begin by reviewing some crucial facts and definitions in abstract algebra, especially free groups, then focus on the Seifert Van-Kampen theorem, which tells us how to compute the fundamental group of a given space if it can be decomposed into constituent open subspaces with nice properties. It will enable us to compute the fundamental groups of a much larger class of spaces. In chapter 4, we will see how the fundamental group and the covering space play roles surprisingly similar to that of field extensions and subgroups of the Galois group in the fundamental theorem of Galois theory. Readers are assumed to be familiar with basic notions in the point-set topology and also abstract algebra, especially in the quotient group and isomorphism theorems. Most materials are introduced in [Mun00b]. i ii PUBO HUANG Acknowledgement I would like to thank my advisor, Professor Monty McGovern, for his lightning-fast reply speed so that I always had my questions cleared up, also for his advice not only on mathematics but other parts of my life. I want to thank Nikou Lei[Lei], for accompanying me through the pandemic, bringing a lot of joy, and encouraging me to think more and deeper about my life. I want to thank Guangqiu Liang and Kevin Kim for their time spent dis- cussing math with me, and Chang Yang for his funny dialect that I listened to when stressed out. Finally, I want to thank my family for supporting me wherever I go. THE FUNDAMENTAL GROUP AND ITS APPLICATIONS iii Contents Introductioni Acknowledgement ii Notations iii 1. Construction of the Fundamental group1 2. Covering Spaces8 3. Seifert-Van Kampen Theorem 14 4. Galois Correspondence of Covering Spaces 22 References 27 Notations Z; C; R: the integers, complex numbers, and real numbers. I = [0; 1]: the unit interval. n R : the n-dimensional Euclidean space. n+1 Sn: the unit sphere in R . F : disjoint union ∼=: isomorphism of groups, homeomorphism ': homotopy. THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 1 1. Construction of the Fundamental group In this chapter, we will prepare ourselves with some basic notions that are fundamental (no pun intended) to the construction of the fundamental group. Standing Assumption: For the entire thesis, any function or map men- tioned is assumed to be continuous unless otherwise mentioned. Definition 1.1. Let X be a space. A path is a (continuous) function f : I ! X, where I = [0; 1] is the unit interval. In particular, the constant map t 7! c for all t 2 I and some fixed c 2 X is also a path. Definition 1.2. If f and f 0 are maps from the space X into the space Y , we say that f is homotopic to f 0 if there is a map F : X × I ! Y satisfying F (x; 0) = f(x) and F (x; 1) = f 0(x) for each x. F is consequently called a homotopy between f and f 0. In symbols we write f ' f 0. If f ' c and c is the constant map, then f is nulhomotopic Definition 1.3. If f and f 0 are two paths in X with the same initial point x0 and final point x1, and there is a map H : I × I ! X such that H(t; 0) = f(t) and H(t; 1) = f 0(t); H(0; s) = x0 and H(1; s) = x1; for all (s; t) 2 I × I. We say that H is a path homotopy between f and f 0, 0 and we write f 'p f : The definition of path homotopy captures our intuition of \continuously deforming" a path to another path. We can think of t as representing the time, and s as a continuous label for a particular path. For example, the first line of equations says that the \0-th" function is our function f, and the \1-st" function is the function f 0: Moreover, the second line of equations tells us that for any \s-th" path, it always starts at x0 and ends at x1: One thing to notice is that the entire Figure 1 is a path homotopy, which consists of a collection of paths. For this reason, we may also think of H as a one-parameter family of paths. Figure 1. Path Homotopy 2 PUBO HUANG Notation 1.4. If f is a map that maps the point xn 2 X to point yn 2 Y respectively, we denote this fact by f :(X; x1; x2; :::) ! (Y; y1; y2; :::): Theorem 1.5. Path homotopy is an equivalence relation. Fix a space X, we check that the three properties of the equivalence relation are satisfied by 'p. Proof. (1) Reflexivity: Given a path f :(I; 0; 1) ! (X; x0; x1); then H : I × I ! X defined by H(t; s) = f(t) is the desired path homotopy. 0 (2) Symmetry: If f 'p f via the path homotopy H; then the map H0 : I × I ! X defined by H0(t; s) = H(t; 1 − s) does the job since H0(t; 0) = H(t; 1) = f 0(t) and H0(t; 1) = H(t; 0) = f(t): (3) Transitivity: Suppose that f 'p g and g 'p h via the path homotopies 0 H and H , respectively. If f is such that f :(I; 0; 1) ! (X; x0; x1) then so is g. The same holds for g and h: In particular, the three paths start at x0 and end at x1: We construct the homotopy G between f and h by \gluing" H and H0. Define G : I × I ! X by ( H(t; 2s) if s 2 [0; 1 ] G(t; s) = 2 0 1 H (t; 2s − 1) if s 2 [ 2 ; 1]: 1 0 1 Note that since H(t; 2 ) = g(t) = H (t; 2 ) and by pasting lemma, G is continuous, and is our desired path homotopy. Thus, 'p partitions X into disjoint union of equivalence classes, which we denote by [f]. [f] is the set of all paths that are path homotopic to f: If we have two paths f and g such that f(1) = g(0), then we can form a new path f ∗g, called the \product path" of f and g, by traversing at twice the speed of f; g. Namely, we define ( f(2t); t 2 [0; 1 ] (f ∗ g)(t) = 2 1 g(2t − 1); t 2 [ 2 ; 1]: Again h is continuous by the pasting lemma, and so h is a path. THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 3 An important observation is the following, Lemma 1.6. Product operation ∗ respects homotopy classes. That is, if we take any f 0 2 [f] and g0 2 [g] such that f 0(1) = g0(0), we have that 0 0 f ∗ g 'p f ∗ g: Proof. First note that since f 0 ' f and g0 ' g, it is the case that f(1) = f 0(1) = g0(0) = g(0): Let F be the homotopy between f; f 0 and G the homo- topy between g; g0. Consider the map H defined by ( F (2t; s); t 2 [0; 1 ] H(t; s) = 2 1 G(2t − 1; s); t 2 [ 2 ; 1]: H is continuous because F; G are continuous and F (1; s) = f 0(1) = g0(0) = G(0; s): Hence H is a path homotopy between f 0 ∗ g0 and f ∗ g since ( F (2t; 0) = f(2t) H(t; 0) = G(2t − 1; 0) = g(2t − 1); which is exactly f ∗ g; and ( F (2t; 1) = f 0(2t) H(t; 1) = G(2t; 1) = g0(2t); which is exactly f 0 ∗ g0: This implies that we may define an operation on the equivalence classes of paths (as long as the two paths are such that f(1) = g(0)) by [f] ∗ [g] = [f ∗ g]: Figure 2. Homotopy of the product Definition 1.7. Let X be a space. A loop is a path such that f(0) = f(1): If f(0) = f(1) = x0, we say that f is a loop based at x0: Definition 1.8. Given any path f : I ! X and γ : I ! I such that γ(0) = 0 and γ(1) = 1, the composition f ◦γ is a reparametrization of f. Intuitively, f ◦ γ simply traverses the image of f with different speed (but in the same direction as f!). 4 PUBO HUANG Lemma 1.9. Given a path f : I ! X and its reparametrization f ◦ γ, there is a homotopy H so that f ' f ◦ γ. Proof. Notice that there is a homotopy between γ and Id : I ! I in I via H(t; s) = (1 − s)γ(t) + st: We check that H is indeed a map from I × I onto I. Note that we have the equivalent conditions (1 − s)γ(t) + st ≥ 0 () (s − 1)γ(t) ≤ st; which always holds since s − 1 ≤ 0 and γ(t) 2 [0; 1] and s; t are non-negative.