The Fundamental Group and Its Applications

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 mentioned 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 ∈ I and some fixed c ∈ 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) ∈ 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 ∈ X to point yn ∈ 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 ∈ [0, 1 ] G(t, s) = 2 0 1 H (t, 2s − 1) if s ∈ [ 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 ∈ [0, 1 ] (f ∗ g)(t) = 2 1 g(2t − 1), t ∈ [ 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 ∈ [f] and g0 ∈ [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 homotopy between g, g0. Consider the map H deﬁned by ( F (2t, s), t ∈ [0, 1 ] H(t, s) = 2 1 G(2t − 1, s), t ∈ [ 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) ∈ [0, 1] and s, t are non-negative. For any t, either γ(t) ≥ t or γ < t. In the ﬁrst case, (1 − s)γ(t) + st ≤ (1 − s)γ(t) + sγ(t) = 1.

Figure 3. Straight-line homotopy

Entirely similar argument for when γ < t shows that H(t, s) ≤ 1. Hence H is a homotopy between γ and Id. Another way to see this is that any two paths in the unit square (which is convex) can always be continuously transformed to each other via the straight-line homotopy. Now, f ◦ H is a path homotopy between f = f ◦ Id and f ◦ γ since f ◦ H(t, 0) = f(γ(t)), and f ◦ H(t, 1) = f(t), and f ◦ H(0, s) = (1 − s)γ(0) + 0 = 0, and f ◦ H(1, s) = (1 − s)γ(1) + s = 1.

Hence f ' f ◦ γ. With these important observations, we are now ready to put an algebraic structure on our space. THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 5

Theorem 1.10. Let X be a space, and x0 ∈ X be ﬁxed. Then the set of all path equivalence classes based at x0 forms a group under the operation ∗. We call this group “the fundamental group of X based at x0” and denote it by π1(X, x0). Proof. We check that the axioms of a group are satisﬁed.

(1) Closure: This is freely given since the product of two loops at x0 is still a loop at x0. (2) Associativity: If f, g, h are paths such that f(1) = g(0) and g(1) = h(0), then (f ∗ g) ∗ h and f ∗ (g ∗ h) are defined. Let k = (f ∗ g) ∗ h, then f ∗ (g ∗ h) have the same image as k, f ∗ (g ∗ h)(0) = k(0), and f ∗ (g ∗ h)(1) = k(1) by definition of ∗. One can check that f ∗ (g ∗ h) is a reparametrization of k. So we can find some γ such that k ◦ γ = f ∗ (g ∗ h). By Lemma 1.9, we have that k = (f ∗ g) ∗ h ' f ∗ (g ∗ h), which implies [f ∗ g] ∗ [h] := [(f ∗ g) ∗ h)] = [f ∗ (g ∗ h)] =: [f] ∗ [g ∗ h]. Hence the operation ∗ is associative. In particular, the result holds for loops.

(3) Identity: Consider the constant loop x0, i.e., the loop c(t) = x0 for t ∈ [0, 1]. Given any loop f based at x0, it suﬃces to show that f ∗ x0 ' f and f ' x0 ∗ f. But f ∗ x0 is just traversing the image of f from time 1 [0, 2 ], then remain at x0 for the rest of the time. It is easy to see that f ∗ x0 is a reparametrization of f, and by Lemma 1.9, f ' f ∗ x0. Replacing f ∗ x0 with x0 ∗ f in the above proof, we have

[x0] ∗ [f] = [x0 ∗ f] = [f] = [f ∗ x0] = [f] ∗ [x0]. ¯ (4) Inverse: For any loop f based at x0, consider the inverse f deﬁned by f¯(t) = f(1 − t), which has same image as f but traversed in the reverse direction. Consider the homotopy as a one-parameter family

hs of paths. We deﬁne hs = fs ∗ gs where ( f(t), t ∈ [0, 1 − s] fs(t) = f(1 − s), t ∈ [1 − s, 1].

That is, fs(t) is traversing, in direction of f, the image of f until t = 1 − s, then remain stationary at the point f(1 − s) for the rest of

the time. Let gs = fs, then hs is the product path in Figure 4. Then h0(t) = f(t) and h1(t) = f(0) = x0. So H(t, s) := hs(t) is a path homotopy between f ∗ f and x0. Since we are simply choosing one path and call it f, we may also call its inverse f So f ∗ f ' x0 as well. Hence

[f] ∗ [f] = [f] ∗ [f] = [x0]. 6 PUBO HUANG

Figure 4. Existence of Inverse

For any π1(X, x0), we can see that π1(X, x0) = π1(C, x0) if C is the path component containing x0. This is so because for any two loops f, g based at x0, f ∗g is a path-connected subspace of X, and so they can not lie in two path components. Therefore, we might assume, without loss of generality, that X is path-connected. A natural question to ask is: what is the relationship between

π1(X, x0) and π1(X, x1)? To answer this, we need to assume that x0, x1 lie in the same path component. Since x0, x1 lie in the same path component, there is a path α such that α(0) = x0 and α(1) = x1. For each loop f based at x0, we can associate it with a loop α ∗ f ∗ α, which is based at x1. Because ∗ is associative on the equivalence classes, [α ∗ f ∗ α] = [α] ∗ [f] ∗ [α]. We get a mapα ˆ : π1(X, x0) → π1(X, x1), deﬁned by αˆ([f]) = [α] ∗ [f] ∗ [α]. This map is well deﬁned because ∗ respects homotopy classes.

Theorem 1.11. If X is path-connected,α ˆ is an isomorphism between π1(X, x0) and π1(X, x1)

Proof. We show thatα ˆ is a bijective homomorphism. for any [f], [g] ∈ π1(X, x0), we have αˆ([f] ∗ [g]) =α ˆ([f ∗ g]) = [α] ∗ [f ∗ g] ∗ [α] = [α] ∗ [f] ∗ [g] ∗ [α] (1) = [α] ∗ [f] ∗ [α] ∗ [α] ∗ [g] ∗ [α] =α ˆ([f]) ∗ αˆ([g]).

If we let αˆ([f]) = [α] ∗ [f] ∗ [α], then αˆ ◦ αˆ([f]) =α ˆ([α] ∗ [f] ∗ [α]) (2) = [α] ∗ [α] ∗ [f] ∗ [α] ∗ [α] = [f].

Entirely similar computation shows that αˆ is the two-sided inverse ofα ˆ. Soα ˆ is bijective. THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 7

Remark 1.12. Therefore, if X is path-connected, the structures of π1(X, x) are all the same for each x ∈ X. But there is one caveat. Because there can be multiple paths from x0 to x1 and the isomorphism above relies on a speciﬁc path, there is no natural (canonical) map to identify elements of π1(X, x0) with elements of π1(X, x1). For this thesis, however, it suﬃces to just think of “the” fundamental group of X, without reference to the base point, and we will omit the base point if no confusion arises.

Deﬁnition 1.13. It might happen that π1(X, x0) is the trivial group, that is, π1(X, x0) = {[x0]}. In this case, we say that X is simply connected.

If X is simply connected, then any loop at x0 can be continuously “shrunk” to a point.

n Example 1.14. The Euclidean space R is simply connected. To show this we need to show that every loop is nulhomotopic. Let x be the base point. n Then for any loop f we may deﬁne H : I × I → R by H(t, s) = (1 − s)f(t) + sx. This is an example of the straight-line homotopy illustrated in Figure3. 8 PUBO HUANG

2. Covering Spaces Having deﬁned the fundamental group, it is natural to ask what the fundamental groups of various spaces look like. Perhaps the unit circle S1 is among one of them. If the whole space is the unit circle, our daily experience tells us that a rope in a shape of a circle cannot be torn apart and shrink to a point continuously and that a rope in the shape of an arc can. So it is tempting to 1 guess that π1(S , x0) is isomorphic to the integers since the equivalence class of the paths is paths that wrap around a circle an integer amount of times.

In this chapter, we will focus on the concept of covering space, which will 1 ∼ confirm our guess that π1(S ) = Z and see in later chapter exactly how covering spaces and the fundamental group presents us with a well-choreographed dance. For now, we introduce the definition and some theorems. Definition 2.1. If E and B are topological spaces and p : E → B is surjective. We say that an open set U of B is evenly covered by p if the preimage −1 p (U) is the union of disjoint open sets Vα in E and that for each α, Vα is homeomorphic to B via the restriction p|Vα . It’s customary to draw the following picture of “stacks of pancakes” to indicate this idea.

Figure 5. Stacked pancake

Deﬁnition 2.2. Let p : E → B be surjective. If every b ∈ B has a neighborhood U that is evenly covered by p, then we say that p is a covering map, and E is the covering space.

Theorem 2.3. The map p : R → S1 defined by p(x) = (cos 2πx, sin 2πx) is a covering map. Intuitively, we can imagine the real line as an infinite helix above S1, and p squashes [n, n + 1), n ∈ Z, onto S1. Proof. Since any open set in S1 is the union of open arcs, it suffices to consider an arbitrary open arc A. We can characterize A by determining its starting and ending angles, α, β such that α < β. THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 9

Figure 6. An arc is determined by two angles

S α β The preimage of this arc under p is the union k( 2π + k, 2π + k) where k ∈ Z. Each such interval is homeomorphic to the arc and since β − α < 2π, β α the endpoint 2π + k of one interval doesn’t touch the starting point 2π + k + 1 of the next , so the union is disjoint. Deﬁnition 2.4. Let p : E → B be a map. If f is a mapping of some space X into B, a lifting of f is a map f˜ : X → E so that p ◦ f˜ = f. In other words, f˜ makes the following diagram commute. E f˜ p f X B Lemma 2.5. (Lebesgue Number Lemma) Let A be an open covering of the metric space (X, d). If X is compact, there is a δ > 0 such that for each subset of X having diameter less than δ, there exists an element of A containing it.[Mun00a] Deﬁnition 2.6. The diameter of a set A in a metric space (X, d) is the number diam(A) = sup{d(a1, a2): a1, a2 ∈ A}. Remark 2.7. In particular, if I is an interval with endpoints a < b, then diam(I) = b − a.

Lemma 2.8. Let {Vα} be an open cover of B; let f : [0, 1] → B be a path, then there is a subdivision 0 = s0 < s1 < ··· < sn = 1 of [0, 1] such that for each i there is some α such that f([si, si+1]) ⊂ Vα. Proof. We use the Lebesgue Number Lemma. Given a loop f, we ﬁrst show that there is a subdivision x0 < ··· < xn of [0, 1] so that f([xi, xi+1]) is −1 S S −1 contained in some Vα. Observe that f ( Vα) = f (Vα) = [0, 1], and S −1 since Vα is open and f is continuous, f (Vα) is an open cover for [0, 1]. Because [0, 1] is compact, Lebesgue Number Lemma asserts the existence of a

δ > 0 such that for each subset of X, in particular intervals Ix with diameter −1 less than δ, we have Ix ⊂ f (Vα) for some α. It follows that f(Ix) ⊂ Vα. Moreover, the collection of open intervals with diameter less than δ is an open cover for [0, 1]. By compactness, again, there is a ﬁnite subcover A =

{[0, a1), ..., (ai, bi), ..., (an, 1]} that covers [0, 1]. Index them properly so that 10 PUBO HUANG bi ≥ ai+1 > ai. If bi = ai+1, we leave them as is. If bi > ai+1 we remove {(ai, bi), (ai+1, bi+1)} from A and throw in {(ai, ai+1), (ai+1, bi), (bi, bi+1)}. This process will stop as the number of sets involved is ﬁnite. Since the three new intervals are subsets of the two old intervals, their images under f still lie in Vα. Moreover, the images of their closures also lie in only Vα since the diameters are the same. Hence these ai’s and bi’s deﬁne the desired partition. Lemma 2.9. (Path lifting lemma) Let p : E → B be a covering map, let p(e0) = b0. Any path f : [0, 1] → B beginning at b0 has a unique lifting to a ˜ path f in E beginning at e0.

Proof. Cover B by open sets Uα. Then every Uα is evenly covered by p as it is a covering map. By Lemma 2.8. we can ﬁnd a subdivision 0 = s0 < s1 < ··· < sn = 1 of [0, 1] so that for each i, f([si, si+1]) lies in Uα for some α. We deﬁne the lift f˜ piecewisely.

˜ First, we define f(s) for 0 ≤ s ≤ s1 as follows: f([0, s1]) lies in some U0 of −1 B. The preimage p (U0) is a collection of disjoint open sets Vβ in E each of which is homeomorphic with Uα via p. Then there is some V0 such that ˜ −1 ˜ f(0) = e0 ∈ V0. We define f(s) = p |V0 (f(s)) for 0 ≤ s ≤ s1. Then f will −1 be continuous on [0, s1] as p |U0 : U0 → V0 is a homeomorphism. Now let ˜ s1 play the same role as 0 and we can define f on [s1, s2]. continue in this fashion until f˜ is defined on all of [0, 1]. The continuity of f˜ follows from the pasting lemma (the reason being that if f(si) lies in two open sets Ui and −1 ˜ Ui+1, then p |Ui∩Ui+1 is also a homeomorphism). By definition of f, we see that p ◦ f˜(s) = p ◦ (p−1(f(s))) = f(s), so f˜ is a lifting of f along p. Moreover, such a lifting is unique because once f˜(0) is decided, the later values are all defined using the inverse of a homeomorphism, which is bijective, and hence unique. We can generalize the previous lemma to show that path homotopy can be lifted as well.

Lemma 2.10. Let p : E → B be a covering map with p(e0) = b0. Let ˜ F : I×I → B with F (0, 0) = b0. There is a unique lifting of F to F : I×I → E ˜ such that F (0, 0) = e0. In particular, path homotopy can be lifted. We shall skip the proof here as it is quite similar to Lemma 2.9 but rather tedious. From now on we denote homotopies or paths in the covering space with a tilde ( like f˜) and its image under p without. Equipped with these two 1 lemmas, we are ready to compute π1(S ). Theorem 2.11. The fundamental group of S1 is inﬁnitely cyclic and generated by the homotopy class of the loop `(t) = (cos 2πt, sin 2πt) based at (1, 0). 1 ∼ That is, π1(S ) = Z.

Proof. Let p be as in Theorem 2.3 and let p(0) = x0 = (1, 0) be the base point. Let f be a loop based at (1, 0) and `n(t) = (cos 2nπt, sin 2nπt), which THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 11 is the loop that goes around n times S1. By Lemma 2.9, there is a unique path lifting f˜ beginning at 0. Now, since p ◦ f˜(1) = f(1) = (1, 0), we know that (cos 2πf(1), sin 2πf(1)) = (1, 0). Hence ( cos 2πf(1)˜ = 1 ⇒ 2πf(1)˜ = 2πn sin 2πf(1)˜ = 0 ⇒ 2πf(1)˜ = πk ˜ for n, k ∈ Z . Because f(1) must be an integer to satisfy the ﬁrst equation, we ˜ conclude that f(1) ∈ Z. Hence f˜ is a path from 0 to n for some n ∈ Z. Notice ˜ ˜ ˜ ˜ that `n is also a path from 0 to n. Let H(t, s) = `n(t)(1 − s) + f(t)s. Then ˜ ˜ ˜ ˜ ˜ H is a straight line homotopy in R between `n and f and f ' `n. Applying ˜ p, we have [p ◦ f] = [f] = [`n].

It remains to show that n is uniquely determined by f. Suppose now f ' `n and f ' `m. By transitivity `n ' `m. Let H be the path homotopy between them. Then by Lemma 2.10, H lifts to a unique path homotopy H˜ between ˜ ˜ `n and `m. By deﬁnition of a path homotopy, these two paths must have the ˜ ˜ same starting and ending points. Because `n ends at n and `m ends at m, we must have n = m.

Finally, let `n denote the n-fold product ` ∗ ` ∗ · · · ∗ `. Then `n([0, 1]) = 1 n S = `n([0, 1]) and ` , `n both are loops at (1, 0), so their starting and ending n points agree. We can check that ` is a reparametrization of `n, which implies n n n n `n ' ` and [`n] = [` ]. By induction and definition of ∗, we have [` ] = [`] . 1 n 1 Hence every element in π1(S ) is of form [`] . That is, π1(S ) is cyclic of infinite order, and a basic result in the group theory shows that if G is a ∼ cyclic group with infinite order, then G = Z. ˜ ˜ In the above proof, we showed that `n and `m end at the same point. This is true generally.

Definition 2.12. Let p : E → B be a covering map; let b0 ∈ B. Choose e0 so ˜ ˜ that p(e0) = b0. For [f] ∈ π1(B, b0), f is its lifting with f(0) = e0. We define −1 the lifting correspondence φ : π1(B, b0) → p (b0) of p by φ([f]) = f˜(1). This is a well-defined map since if f 0 ∈ [f], then f 0 ' f and there is a path homotopy H between then. We can lift H to a path homotopy H˜ between f˜0 and f˜. So f˜0 and f˜ have the same endpoints.

Theorem 2.13. Let p : E → B be a covering map; let p(e0) = b0. If E is path-connected, the lifting correspondence φ is surjective. If π1(E) = 0, φ is bijective.

−1 ˜ Proof. Let e ∈ p (b0). Because E is path-connected, there is a path f ˜ between e0 and e. Then f := p ◦ f is a loop based at b0. By deﬁnition 12 PUBO HUANG ˜ φ([f]) = f(1) = e. Suppose that π1(E) is trivial. Given φ([f]) = φ([g]), we ˜ ˜ have that f(1) =g ˜(1). So f ∗ g˜ is a loop at e0. Because π1(E) = 0, the only ˜ ˜ homotopy class is the constant loop [f ∗ g˜] = [f] ∗ [g˜] = [e0]. Multiply [˜g] on the right gives [f˜] = [˜g]. Hence there is a path homotopy H˜ between f˜ and ˜ g˜. Then H = p ◦ H is a path homotopy between f and g, so [f] = [g]. 1 Having computed π1(S ), we can compute the fundamental group of the torus via the following theorem. Some notations are needed.

Deﬁnition 2.14. Let h :(X, x0) → (Y, y0) be a map. Deﬁne

h∗ : π1(X, x0) → π1(Y, y0) by

h∗([f]) = [h ◦ f].

Then h∗ is a homomorphism of groups and we call h∗ the induced homomorphism by h. Remark 2.15. Induced homomorphism will be used extensively and repeat- edly throughout the thesis. There are two important properties of the induced homomorphisms. If h :(X, x0) → (Y, y0) and k :(Y, y0) → (Z, z0), then

(1)( k ◦ h)∗ = k∗ ◦ h∗ (2) i∗ : π1(X) → π1(X) is the identity if i = Id : (X, x0) → (X, x0). ∼ (3) If (X, x0) '(Y, y0) via h, then π1(X) = π1(Y ). The prove the last claim simply notice that the inverse h−1 induces a group −1 −1 −1 −1 homomorphism h∗ . Using (1), we have (h ◦ h )∗ = h∗ ◦ h∗ = i∗, so h∗ is the inverse for h∗, which shows that h∗ is a bijective homomorphism, hence an isomorphism. ∼ Theorem 2.16. If X×Y is path-connected, then π1(X×Y ) = π1(X)×π1(Y ).

Here π1(X) × π1(Y ) is the product group formed by two groups with the understood group operation “ · ”, ([f] × [g]) · ([f 0] × [g0]) = [f ∗ f 0] × [g ∗ g0].

We assume, of course, that x0, y0 are the base points for X,Y , respectively, and x0 × y0 the base point for X × Y. Proof. Let

p :(X × Y, x0 × y0) → (X, x0) and

q :(X × Y, x0 × y0) → (Y, y0) be the projections. Then we have the induced homomorphisms p∗ and q∗. We may deﬁne the product homomorphism

Φ: π1(X × Y ) → π1(X) × π1(Y ) via

Φ([f]) = p∗([f]) × q∗([f]) = [p ◦ f] × [q ◦ f]. THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 13

It suﬃces to show that Φ is bijective.

Given [fX ] × [fY ] ∈ π1(X) × π1(Y ), deﬁne f : I → X × Y by

f(t) = fX (t) × fY (t).

Then by deﬁnition Φ([f]) = [p ◦ f] × [q ◦ f] = [fX ] × [fY ]. So Φ is surjective.

Suppose that [f] ∈ ker Φ, then Φ([f]) = [p◦f]×[q ◦f] is the identity. Hence p ◦ f 'p x0 and q ◦ f 'p y0. Let F,G be their respective path homotopies, we have H : I × I → X × Y deﬁned by H = F × G is a path homotopy between f and x0 × y0. Thus Φ is injective. Example 2.17. The torus T ∼= S1 × S1 has fundamental group isomorphic 1 1 ∼ to π1(S ) × π1(S ) = Z × Z. 2 Example 2.18. The solid torus S1 × B1, where B1 is the solid disk in R has 1 1 ∼ 1 1 ∼ ∼ fundamental group π1(S × B ) = π1(S ) × π1(B ) = Z ×e = Z. The fact that B1 is simply connected is by the existence of a straight-line homotopy in the 2 convex set B1 ⊂ R . 2 ∼ 1 Example 2.19. The punctured plane R −{0} = S × R has fundamental 1 ∼ ∼ group isomorphic to π1(S ) × π1(R) = Z ×{e} = Z . 14 PUBO HUANG

3. Seifert-Van Kampen Theorem In this section, we will meet the heavy machinery that helps us enlarge significantly the number of fundamental groups we can compute. This is the Seifert Van Kampen’s theorem. According to Van Kampen himself, the mo- tivation for this theorem is that “the opportunity of simplifying the treatment of a fundamental group by means of this theorem has been overlooked several times.... For this reason we do not think it superfluous to devote a separate paper to it.” [Kam33] The role this theorem plays is like that of semiconductors, after the invention of which we had an explosion in our computational power. We will see its simple version first and introduce some crucial concepts in group theory in preparation for the classical Seifert-Van Kampen’s Theorem. Definition 3.1. Given a group G and a subset A ⊂ G, we say that A generates G if every element g ∈ G can be written as a finite product of elements in A.

Example 3.2. The integers Z is generated by {1}. Example 3.3. The even numbers {..., −4, −2, 0, 2, 4, ...} are generated by {1}. This is easily seen to be the case since the set of even numbers is a subset of the integers. Note that, however, this does not say {1} only generates the even numbers, since it also generates Z . Theorem 3.4. If a space X is the union of a collection of path-connected open sets Aα each containing the basepoint x0 ∈ X and if each intersection Aα ∩ Aβ is path-connected, then every loop f in X at x0 is homotopic to a product of loops each of which is contained in a single Aα. If X = U ∪ V , then this lemma says that any loop in X is homotopic to a product of loops each of which is contained in either U or V. In Munkres’ book, this is stated diﬀerently in terms of fundamental groups and induced homomorphisms. Theorem 3.5. Suppose X = U ∪V , where U, V are open in X. Suppose that

U ∩ V is path-connected, and that x0 ∈ U ∩ V. Let i and j be the inclusion mappings of U and V , respectively, into X. Then the images of the induced homomorphisms

i∗ : π1(U, x0) → π1(X, x0) and j∗ : π1(V, x0) → π1(X, x0) generate π1(X, x0). Note that these two theorems say the same thing. Suppose X = U ∪ V as in Theorem 3.4. If Theorem 3.4 is true, then any loop f in X is homotopic to a product of loops each of which is contained in either U or V . Passing the loop to its path-homotopic class in the fundamental group, this says that [f] is the product of homotopy classes of loops each of which is contained in the fundamental group of U or V . Viewing these component loops as THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 15 loops in π1(U) or π1(V ) and apply i∗, j∗, we have Theorem 3.5. Conversely, if [f] ∈ π1(X), then [f] is the product [g1] ∗ [g2] ∗ · · · [gn], where we just view [gi] as an element in either π1(U) or π1(V ).

Proof. By Theorem 2.8, we can find a partition a0 < ··· < an of the unit interval such that f([ai−1, ai]) lies in only U or V . Moreover, this partition can be made so that f(ai) ∈ U ∩ V. If f(ai) ∈ U ∩ V for all i, then there is nothing to do. If not, we find the first index i such that f(ai) 6∈ U ∩ V and assume without loss that f(ai) ∈ U. Observe that this forces f([ai−1, ai]) and f([ai, ai+1]) to lie both in U. We can then merge these two partitions to [ai−1, ai+1] and the image of this new interval still lies in U. Do this to every partition point and we obtain a new partition of I such that each partition point’s image under f lies in U ∩ V.

Figure 7. Each loop is contained in only U or V .

Having chosen a partition a0 < ··· < an with the desired property, we consider the restricted path f|[ai−1,ai]. Let fi to be its reparametrization so that fi is deﬁned on [0, 1]. Then f ' f1 ∗ f2 ∗ · · · fn by Lemma 1.9 and the observation that they are reparametrizations of each other. Since each fi(1) = f(ai) ∈ U ∩ V , we can ﬁnd a path gi connecting x0 with f(ai) that lies only in U ∩ V . Thus

(f1 ∗ g1) ∗ (g1 ∗ f2 ∗ g2) ∗ · · · ∗ (gn−1 ∗ fn) ' f is the product of loops (indicated by parentheses) each of which is in either U or V . Passing the above equation to the fundamental group, we see

[f1 ∗ g1] ∗ [g1 ∗ f2 ∗ g2] ∗ · · · ∗ [gn−1 ∗ fn] = [f].

Example 3.6. Sn, the n + 1 dimensional unit sphere with n ≥ 2, is simply connected. Proof. Let U = Sn \{N} and V = Sn \{S}, where N = (0, 0, ..., 1) is the north pole and S = (0, 0, ..., −1) is the south pole. Since the punctured n ∼ n sphere S \{x} = R via the stereographic projection, π1(U) = π1(V ) = 0 is trivial. Take a point x0 that lies on the equator, then x0 ∈ U ∩ V and U ∩ V = Sn \{N,S} is path-connected. By Theorem 3.4, any loop in Sn is 16 PUBO HUANG homotopic to product of loops each of which is contained in either Sn \{N} or Sn \{S}. But as the two latter spaces are simply connected, any loop in n n S is nulhomotopic. Thus π1(S ) = 0.

Lemma 3.7. Given ﬁnitely distinct points F = {(p11, ..., p1m), ..., (pn1, ..., pnm)} m in R , there exists a hyperplane P such that the points in F lie on both sides of P.

Proof. Denote the set of numbers of i-th coordinates from F by Fi = {p1i, ..., pni}, Let M = min Fm, i.e., the smallest m-th coordinate. Let L be the hyperplane 2 xm = M. We consider R : In this case L is a line. Two cases are possible: (1) The entire F lies on L: Simply let P to be a line perpendicular to L

with min F1 < x < max F1. Then we can see that P has the desired property. (2) F does not lie entirely on L : Find > 0 such that L + is above the point with y-coordinate m and below all other points.

Figure 8. Case 1 Figure 9. Case 2

m−1 m Suppose inductively that this holds for R . Consider R . Two cases are possible: (1) F lies entirely on L:. In this case L is a subspace homeomorphic to m−1 R , so the inductive hypothesis applies. (2) F does not lie entirely on L : We can find > 0 so that L + is above the point with last coordinate m and below all other points. This completes the proof. n Example 3.8. If n ≥ 3 and F ⊂ R is finite, then F c is simply connected. n Proof. If F = {x}, then R −{x} is homeomorphic with Sn−1 × R. This can be seen by the fact that the conversion between Cartesian coordinate and the n polar coordinate C : R −{0} → Sn−1 × (0, ∞) defined by x x 7→ , ||x|| ||x|| n ∼ n ∼ is continuous. Moreover, R −{0} = R −{x} and R = (0, ∞). By Theorem n ∼ n−1 n−1 ∼ 2.16, R −{x} = S ×R has trivial fundamental group since π1(S ×R) = n−1 ∼ ∼ π1(S )×π1(R) = {e}×{e} = {e}. Now suppose that for |F | = 1, 2, ...k −1, n R −F is simply connected. Consider the case when |F | = k. By Lemma 3.7, THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 17

n we can find a hyperplane L = {(x1, ..., xn) ∈ R : xn = t} so that it splits these k points. Define − n + n L {(x1, ...xn) ∈ R : xn < t + } and L = {(x1, ...xn) ∈ R : xn > t − }, where is chosen so that L− and L+ intersect and include the points split by L, respectively (this is feasible as we only have finitely many points, so we n may just wiggle the plane a little bit). Then R −F = L− ∪ L+, and we can n apply Theorem 3.4 to see that π1(R −F ) is genearted by the trivial group, which is trivial.

Figure 10. We can wiggle a little bit!

Corollary 3.9. If X = U ∪ V as in Theorem 3.5 and U, V are simply connected, then X is simply connected. We now turn our attention to the Seifert-van Kampen Theorem. Recall that

Theorem 3.5 tells us the images of the induced homomorphisms i∗ : π1(U) → π1(X) and j∗ : π1(U) → π1(X) generate π1(X). In fact, Seifert-Van Kampen theorem will say that they completely characterize π1(X). To see this, we need some abstract algebra. Definition 3.10. Let G, H be groups (if they are isomorphic, we use different sets of notations so that they are distinct groups), then the free product G ∗ H is defined to be the set of strings of formal letters (called words)

g1h1g2h2...gnhn, where gi ∈ G and hi ∈ H. The operation on G ∗ H is the concatenation of strings, and the only relation on the elements are gieH gi+1 = (gigi+1) ∈ G and higH hi+1 = (hihi+1) ∈ H, i.e., we can throw away the identities in G or H if they appear in a word, and replace two consecutive letters in the same group with their product in either G or H. Thus, for any formal letters described above, we can apply these two relations so that after ﬁnitely many steps we obtain a word that we cannot apply these two relations again, and we call them the reduced words. 18 PUBO HUANG

Remark 3.11. We will treat G, H as subgroups of G ∗ H by taking the inclusion homomorphism G,→ G ∗ H and H,→ G ∗ H.

Remark 3.12. Given homomorphisms φG : G → A and φH : H → A for an arbitrary group A, we can extend them to a unique homomorphism Φ : G ∗ H → A by requiring that

Φ(g1h1g2h2...gnhn) = φG(g1)φH (h1)...φG(gn)φH (hn).

Theorem 3.13. Let X = U ∪V , where U and V are open in X; assume U, V , and U ∩ V are path-connected; let x0 ∈ U ∩ V . Let H be a group, and let

φ1 : π1(U) → H and φ2 : π1(V ) → H be homomorphisms. Let i1, i2, j1, j2 be the homomorphisms indicated in the following diagram, each induced by inclusion.

If φ1 ◦ i1 = φ2 ◦ i2, then there is a unique homomorphism Φ : π1(X) → H such that Φ ◦ j1 = φ1 and Φ ◦ j2 = φ2.

i1 π1(U ∩ V ) π1(U)

i2 j1 φ1 j2 π1(V ) π1(X) Φ

φ2 G

This is the modern version of the Seifert-Van Kampen Theorem. The above diagram is the same diagram for a “pushout” in the category theory. Inter- ested readers can ﬁnd more in [nla20]. We shall not present the proof here as it is quite involved. A constructive and technical proof can be found in [Mun00b], and an “one-line” proof that relies on covering space theory (with the assumption that U, V and U ∩V all have simply connected covering spaces) is due to Grothendieck [Ful07]. We prove the classical version.

Remark 3.14. The next proof will be a simpliﬁcation of the proof in [Hat00b], S which deals with the general case that X = Aα. But very often we use Seifert-Van Kampen when the space is just the union of two open subspace.

Theorem 3.15. (Seifert-Van Kampen Theorem) Let jU : π1(U) → π1(X) and jV : π1(V ) → π1(X) be the inclusion-induced homomorphisms. By Remark 3.12, they extend to a unique homomorphism Φ : π1(U)∗π1(V ) → π1(X). Let iU : π1(U ∩ V ) → π1(U) and iV : π1(U ∩ V ) → π1(V ) be the inclusion-induced homomorphisms and N the normal subgroup containing all words of the form −1 −1 −1 iU (`) iV (`) (hence containing also their conjugates wiU (`) iV (`)w for all w ∈ π1(U) ∗ π1(V )) for all ` ∈ π1(U ∩ V ). Then there is an isomorphism π (U) ∗ π (V ) 1 1 ∼= π (X). N 1 THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 19

Proof. In Theorem 3.5, we already saw that the images of jU and jV genearte π1(X). In particular, any loop in X can be written as product of loops in U or V , then taking this product as a word in the free product π1(U)∗π1(V ), we see that Φ is surjective. By the ﬁrst isomorphism theorem, it suﬃces to show that ker Φ = N. We will introduce some terminologies. By a factorization of

[f] ∈ π1(X), we mean a formal product [f1] ··· [fk] (observe the lack of “ ∗ ” here since we are treating them as words) such that:

• Each fi is a loop in U or V , and [fi] ∈ π1(U) or π1(V ) is the homotopy class of fi. • f ' f1 ∗ · · · fk. With the above deﬁnition, a factorization of [f] is a word in the free product

π1(U) ∗ π1(V ), potentially unreduced, i.e., it is possible that fi−1, fi are both loops in U (or V.)

Call two factorizations of [f] equivalent if we can get to one from the other via a ﬁnite number of the following operations or their inverses:

• Write [fi−1][fi] as [fi−1 ∗ fi] if fi−1, fi ∈ π1(U) (or π1(V )). • View the term [fi] ∈ π1(U) (or π1(V )) as an element in π1(V ) (or π1(U)) if fi ∈ U ∩ V.

For a word [f1] ··· [fk] ∈ π1(U) ∗ π1(V ), the first operation is simply one of the relations defined in Definition 3.10. So this operation fixes the factorization as a word in the free product. Suppose now fi is a loop in U ∩V . We write [fi]U and [fi]V to mean that [fi] is viewed as an element in π1(U) or π1(V ), −1 respectively. Since N contains words of form iU (`) iV (`) for ` ∈ π1(U ∩ V ), −1 it contains [fi]U [fi]V . Since N is normal, Nw = wN for all w ∈ π1(U)∗π1(V ) −1 and [fi]U [fi]V N = N. Hence N commutes with all the [fm]’s, then if we re- gard [fi] as [fi]U , we have, after taking projection of the word to the quotient group,

[f1] ··· [fi]U ··· [fk]N = [f1] ··· [fi]U N ··· [fk] −1 = [f1] ··· [fi]U ([fi]U [fi]V N) ··· [fk] −1 = [f1] ··· ([fi]U [fi]U )[fi]V N ··· [fk]

= [f1] ··· [fi]V ··· [fk]N Therefore, the second operation does not aﬀect the coset associated with the factorization in the quotient group. To summarize, equivalent factorizations give the same element in the quotient group.

It is enough to show that, then, any two factorizations of [f] is equivalent since they are mapped under Φ to the same element in π1(X), and this says 0 0 that Φ is injective, and so ker Φ = N. Let [f1] ··· [fk] and [f1] ··· [fm] be two 0 0 factorizations of [f]. By deﬁnition of the factorization, f1 ∗· · ·∗fk ' f1 ∗· · ·∗fm via the homotopy H : I×I → X. Use Lebesgue number lemma to I×I with the −1 −1 open cover H (U) ∪ H (V ) to ﬁnd δ > 0 such that diam([ti−1, ti] × I) < δ 20 PUBO HUANG

(that is, the diagonal of the rectangle is less than δ). Then [ti−1, ti] × I is mapped to either U or V by H (in fact, each corner of the subrectangle can be made to lie in the intersection U ∩ V under the map H as in the proof of Theorem 3.4). Consider the broken line-segments in the figure that starts from the left edge and separate the first n rectangles from the rest, and define the path pn to be H restricted on the broken line-segments. For example, the line segments [0, t1] × 0, t1 × I, [t1, 1] × 1 separate the first 1 rectangles from the rest, so p1 = H(t, 0)|[0,t1] ∗ H(t1, s) ∗ H(t, 1)|[t1,1]. Similarly, if there are n 0 0 rectangles in total, pn = H(t, 1) = [f1] ∗ · · · ∗ [fm]. For sake of simplicity, we shall consider [t1, t2] × I and show that p1 is equivalent to p2, and the general

Figure 11. proof follows just with minor notational changes. It follows that through a 0 0 chain of equivalent factorizations, we reach from [f1] ··· [fk] to [f1] ··· [fm], so Φ is injective.

Let g1, g2, g3, g4 be the paths in U∩V from H(t1, 0),H(t2, 0),H(t1, 1),H(t2, 1) to the base point x0, respectively. We also deﬁne similar paths from other vertices of the other subrectangles. Adjoining these gi appropriately with H([ti, ti+1] × 0), H([ti, ti+1] × 1) or H(ti × I), we obtain loops in either U or V similar to that in Theorem 3.4. For example, if we follow the broken line- segments [0, t1] × 0 → t1 × I → [t1, t2] × 1 via H from bottom-left to top-right, adjoining gi’s along the way when we are at a vertex of a subrectangle, we obtain a factorization corresponding the the loop H|A, where A is the broken line-segments. In particular, p1 and p2 have identical factorizations except the places where p1 follows the dashed line-segments and p2 follows the yellow line-segments. Let `2, `3,L2 and L4 be the paths obtaind by following the line segments indicated in the ﬁgure. We are intersted in

(3) (g1 ∗ `2 ∗ g3) ∗ (g3 ∗ `3 ∗ g4) '(g1 ∗ `2 ∗ `3 ∗ g4) THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 21 and

(4) (g1 ∗ L2 ∗ g2) ∗ (g2 ∗ L3 ∗ g4) '(g1 ∗ L2 ∗ L3 ∗ g4). Since the subrectangle traced out by the dashed and yellow line-segments is convex, there is a straight-line homotopy F between the dashed and yellow line-segments. Thus H ◦F is a path homotopy in either U or V between `2 ∗`3 and L2 ∗ L3. As factors,

[g1 ∗ `2 ∗ `3 ∗ g4] = [g1 ∗ L2 ∗ L3 ∗ g4].

So p1 and p2 have equivalent factorizations. This special case generalizes so that pi and pi+1 have equivalent factorizations. Moreover, p0 has factorization 0 0 [f1] ··· [fk] while pn = [f1] ∗ · · · ∗ [fm]. Therefore any two factorizations of [f] are equivalent.

Deﬁnition 3.16. Let Xα be a collection of spaces and xα a chosen point in W Xα, respectively. The wedge sum Xα is the quotient of the disjoint union F Xα where we identify all the chosen points.

Example 3.17. The ﬁgure 8 has fundamental group isomorphic to Z ∗ Z.

Figure 12. Figure 8

Using the above deﬁnition, ﬁgure 8 is homeomorphic with S1 W S1, two circles touching at only one point. Being a singleton set, x0 is path-connected. 1 Since π1(S ) = Z, using Seifert-Van Kampen theorem we see that π1(8) = Z ∗ Z . ∼ Example 3.18. Similarly, given V,U we have π1(V ∨ U) = π1(V ) ∗ π1(U). ∼ Corollary 3.19. If U ∩ V is simply connected, π1(X) = π1(U) ∗ π1(V ). Notice that the previous examples are special cases. The intersection U ∩V is a single point. To see that the corollary is true notice that the normal subgroup N is contained in the trivial group π1(U ∩ V ) and so Φ becomes an isomorphism between the free product and π1(X).

Example 3.20. Actually, for any given group G, there is a space XG with ∼ π1(XG) = G [Hat00c]. 22 PUBO HUANG

4. Galois Correspondence of Covering Spaces There are striking similarities between how covering spaces relate to the fundamental groups and how field extensions relate to subgroups of a Galois group. There is simply an aesthetic pleasure when one finds that these two different realms (one geometrical, the other purely algebraic) can have formally identical relationships.

In studying covering spaces, one often restricts themselves to base B that is locally path-connected to allow for the fundamental group to exist (since we need to be able to adjoin paths and connect two points via a path to form the fundamental group). Having assumed local path-connectedness, we might as well assume that B is path-connected since B is the disjoint union of its path components, which are path-connected. But if this is the case, the inverse of a covering map p−1, being a local homeomorphism, tells us that p−1(B) is path-connected. Sop ˜ : p−1(B) → B of p restricted is a covering map as well. Thus, in this chapter, we only consider p : E → B where E and B are both path-connected and locally path-connected.

Remark 4.1. It might seem a bit weird why we require E and B to be both path-connected and locally path-connected. Here is an example that illustrates the signiﬁcance. 2 Consider the comb space K as a subspace of R : [ K = {0} × [0, 1] ∪ {1/n} × [0, 1] ∪ [0, 1] × {0}, for n = 1, 2, 3, .... and it looks like a comb (but inﬁnite!).

This space is path-connected since any two points x, y can be joined via a path that starts at x, goes down to [0, 1] × {0}, traverses through it to the line segment to which y belongs and climbs up to y. However, K is NOT locally path-connected. If we take a neighborhood N of x ∈ {0} × (0, 1] such that N does not contain the origin, then N contains some portion of {1/n} × [0, 1] for n large enough. But no path connects x with any of the THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 23 points y ∈ {1/n} × [0, 1] since any path that connects them must go through the origin. Hence, by assuming that E and B are path-connected and locally path-connected, we avoid this kind of “pathological” space.

path-connectedness is a global property. Given a point x ∈ X and an arbitrary neighborhood N of x, path-connectedness does not tell us how x will connect via a path to a point y ∈ N; all it asserts is the existence of such a path. It might be that they are connected via a path that goes outside N and back to N. However, if X is locally path-connected, then we are certain that x, y can be joined by a path simply inside the little bubble N. In this sense, local path-connectedness is stronger than path-connectedness.

Theorem 4.2. (General lifting correspondence theorem) Let p : E → B be a covering map; let p(e0) = b0.

(1) The homomorphism p∗ : π1(E, e0) → π1(B, b0) is a monomorphism. (2) Let H = p∗(π1(E, e0)). The lifting correspondence ϕ induces an injection −1 Φ: π1(B, b0)/H → p (b0) −1 of the collection of right cosets of H into p (b0), which is a bijection if E is path-connected.

(3) If f is a loop in B based at b0, then [f] ∈ H if and only if f lifts to a loop in E based at e0. (1) says that as a cover, hence a “bigger” space, E has simpler fundamental group than B does and π1(E) can be embedded into π1(B). Proof.

(1) It suﬃces to check that ker p∗ = [e0] since p∗ is a homomorphism ˜ ˜ of groups. Suppose that f is a loop at e0 such that f = p ◦ f is nulhomotopic. Then there is a path homotopy H : I ×I → B between ˜ f and b0. By Lemma 2.10, there is a unique lifting H of H such that ˜ ˜ ˜ H(0, 0) = e0. This implies that H is a path homotopy in E between f and the constant loop e0. (2) The induced map is deﬁned by H[f] 7→ φ([f]). To prove injectivity, we show that if φ([f]) = φ([g]), then H[f] = H[g]. That is, given h ∈ H, we have [h] ∗ [f] = [h0] ∗ [g] for some h0 ∈ H; equivalently, [f] = [h] ∗ [h0] ∗ [g] ⇔ [f] ∈ H[g]. Let f, g be such that φ([f]) = φ([g]), and f,˜ g˜ be their respective

liftings to E beginning at e0. Suppose ﬁrst that [f] ∈ H[g]. Then [f] = [h] ∗ [g] = [h ∗ g] where [h] ∈ p∗(π1(E)). By deﬁnition of the induced homomorphism, [p◦h˜] = [h] for some loop h˜ ∈ E, and p◦h˜ ' h via a path homotopy H in B. We may lift H to H˜ in E so that it is a path homotopy between h˜ and the lifting of h, but this says that the ˜ lifting of h is a loop at e0. Thus we may just take h to be a loop in E with p ◦ h˜ = h. Now h˜ is the lifting of h, then h˜ ∗ g˜ is the lifting 24 PUBO HUANG

of h ∗ g. Furthermore, f ' h ∗ g so that their path homotopy lifting to E will say that f˜(1) = h˜ ∗ g˜(1) =g ˜(1). Hence φ([f]) = φ([g]). Conversely, if φ([f]) = φ([g]), then h˜ := f˜ ∗ g˜ is a loop in E basesd ˜ ˜ ˜ ˜ at e0. By deﬁnition of h,[f] = [h ∗ g˜]. Let H be the path homotopy between them. Then p ◦ H˜ is a path homotopy between f and h ∗ g. So [f] = [h] ∗ [g] = [p ◦ h˜] ∗ [g] ∈ H[g]. (3) This is a corollary of (2). Consider the case when g is constant, then ˜ ˜ [f] ∈ H if and only if the endpoint of f is e0, so f is a loop in E based at e0. Lemma 4.3. (general lifting lemma) Let p : E → B be a covering map; let p(e0) = b0. Let f : Y → B, with f(y0) = b0. Suppose Y is path-connected. ˜ ˜ The map f can be lifted to a map f : Y → E such that f(y0) = e0 if and only if

f∗(π1(Y, y0)) ⊂ p∗(π1(E, e0)). Moreover, if the lifting exists, it is unique. This is a generalization of the lifting Lemma 2.10 so that any continuous f can be lifted. Intersted readers can consult [Mun00c; Hat00d].

Some deﬁnitions are in order. Deﬁnition 4.4. Let p : E → B and p0 : E0 → B be covering maps. Then p and p0 are equivalent if there is a homeomorphism h : E → E0 such that p = p0 ◦h. The homeomphirms h is called an equivalence of covering maps. E h E p p0 B

We now show that the subgroup H0 of π1(B) determines the covering p : E → B uniquely, up to the equivalence deﬁned above.

0 0 0 Theorem 4.5. Let p, p be as in Deﬁnition 4.4; let p(e0) = p (e0) = b0. Then 0 0 there is an equivalence h : E → E such that h(e0) = e0 if and only if the groups 0 0 0 H0 = p∗(π1(E)) and H0 = p∗(π1(E )) are equal. If h exists, it is unique. Proof. Suppose that h exists, then since h is a homeomorphism, it induces 0 an isomorphism h∗. In particular h∗ is surjective, so h∗(π1(E)) = π1(E ). By Remark 2.15 and the fact that p = p0 ◦ h, 0 0 0 0 p∗(π1(E)) = (p ◦ h)∗(π1(E)) = p∗(h∗(π1(E))) = p∗(π1(E )). 0 0 Hence H0 = H0. Conversely, suppose that H0 = H0. Note that p : E → B is continuous with p(e0) = b0. Since E is path-connected and locally path- 0 0 connected, by Theorem 4.3, p can be lifted to h : E → E such thatp ˜(e0) = e0. THE FUNDAMENTAL GROUP AND ITS APPLICATIONS 25

E0 h p0 p E B Hence p0 ◦ h = p. We may interchange the roles of E with E0, then we obtain another lifting k with p ◦ k = p0. E k p p0 E0 B Now we show that there is a homeomorphism E ∼= E0. Consider E E0 k◦h h◦k p p0 p p0 E B E0 B

0 We see that p ◦ (k ◦ h) = p ◦ h = p, so k ◦ h is a lifting of p with p(e0) = e0. By uniqueness of the lifting, k ◦ h = Id : E → E. Similarly h ◦ k = Id : E0 → E0, and this shows that h : E → E0 is a bicontinuous map, hence a homeomorphism. There is a subtle point that requires more attention. Since the theorem is of form “If there is an equivalence h, then...” and its converse. Both statements are conditional and do not assert the ex- 0 istence of h in the ﬁrst place. If there is an equivalence carrying e0 to e0, everything works out as expected. If not, it might be the case that there is 0 an equivalence h such that h(e0) = e1. The following theorem answers this.

Theorem 4.6. Let p : E → B be a covering map. Let e0 and e1 be points of −1 p (b0) and Hi = p∗(π1(E, ei)), then

(1) H0 and H1 are conjugate. (2) given e0 and a subgroup H of π1(B, b0) conjugate to H0, there is a −1 point e1 ∈ p (b0) so that H1 = H. With the above theorem, we can relax Theorem 4.5 a little. Theorem 4.7. Let p : E → B and p0 : E0 → B be covering maps with 0 0 0 p(e0) = p (e0) = b0. Then p and p are equivalent if and only if 0 0 0 0 H0 = p∗(π1(E, e0)) and H0 = p∗(π1(E , e0)) are conjugate in π1(B, b0).

0 0 Proof. If h : E → E is an equivalence, and h(e0) = e1. Theorem 4.5 implies 0 0 0 that H0 = H1, and Theorem 4.6 says that H1 is conjugate to H0. Since conjugacy is an equivalence relation, by transitivity we have the desired result. 0 0 0−1 Conversely, if H0 and H0 are conjugate, by Theorem 4.6, there is e1 ∈ p (b0) 26 PUBO HUANG

0 such that H1 = H0. Theorem 4.5 then gives an equivalence h : E → E with 0 h(e0) = e1. Lemma 4.8. If G is a subgroup of Z, then G = n Z for some positive integer n.

Proof. Suppose G is a subgroup of Z. Since Z is well-ordered, let n ∈ G be the smallest positive element. Let g ∈ G be another element. By the division algorithm, we can write g = nq + r, where 0 ≤ r < n. Since G is a group, nq ∈ G as well, then it follows by the closure of operation that g−nq = r ∈ G. Since r < n, it must be the case that r = 0. Therefore, every element of G has the form nq for some q ∈ Z. This means precisely that G = n Z. Example 4.9. We can classify all path-connected covering spaces of S1. In chapter 2, we see that the map p : R → S1 deﬁned by projecting the helix 1 to S is a covering map. Since R is simply connected, p∗(π1(R)) = p∗({e}) = 1 {e} is trivial in π1(S ). Therefore all simply connected covering spaces are 1 equivalent to R because their corresponding subgroup in π1(S ) is trivial, which is conjugate to each other trivially. Moreover, it can be shown that p : S1 → S1 deﬁned by p(z) = zn, where we treat S1 as a subset of C, is ∼ 1 also a covering. In particular, we are sending the generator 1 of Z = π1(S ) ∼ 1 to its n-fold product (sum) n ∈ Z = π1(S ). Thus as a subgroup in codomain S1 it corresponds to n Z . But since every subgroup of Z is of form n Z by the previous lemma, it implies that every path-connected covering of S1 is equivalent to n Z, for n = 0, 1, 2, 3.... For each covering p : E → B it corresponds to a conjugacy class of subgroups in π1(B, b0). Conversely, two coverings are equivalent if and only if they correspond to the same class. Thus, we have now got an injection from the equivalence classes of coverings for B into the conjugacy classes of subgroups of π1(B, b0). If we restrict our X to be semilocally simply connected, that is, for each x ∈ X, there is a neighborhood U of x such that the inclusion- induced i∗ : π1(U) → π1(X) is trivial, we would obtain a bijection. To show the latter involves constructing such a covering space and topologizing it. Readers can consult [Mun00d]. REFERENCES 27

References [Kam33] Egbert R. Van Kampen. “On the Connection between the Funda- mental Groups of Some Related Spaces”. In: American Journal of Mathematics 55.1 (1933), pp. 261–267. issn: 00029327, 10806377. url: http://www.jstor.org/stable/51000091. [Hat00a] Allen Hatcher. Algebraic topology. Cambridge: Cambridge Univ. Press, 2000. url: https://cds.cern.ch/record/478079. [Hat00b] Allen Hatcher. p.44 Algebraic topology. Cambridge: Cambridge Univ. Press, 2000. url: https://cds.cern.ch/record/478079. [Hat00c] Allen Hatcher. p.52 Algebraic topology. Cambridge: Cambridge Univ. Press, 2000. url: https://cds.cern.ch/record/478079. [Hat00d] Allen Hatcher. p.61 Algebraic topology. Cambridge: Cambridge Univ. Press, 2000. url: https://cds.cern.ch/record/478079. [Mun00a] James R. Munkres. §27. Compact subspaces of the real line. Pren- tice Hall, Inc., 2000. [Mun00b] James R. Munkres. §70. The Seifert-van Kampen Theorem. Pren- tice Hall, Inc., 2000. [Mun00c] James R. Munkres. §79. Equivalence of Covering Space. Prentice Hall, Inc., 2000. [Mun00d] James R. Munkres. §82. Existence of Covering Spaces. Prentice Hall, Inc., 2000. [Ful07] William Fulton. §14 Van Kampen Theorem. Springer, 2007. [nla20] nlab. categorical Van Kampen Theorem. 2020. url: https : / / ncatlab.org/nlab/show/van+Kampen+theorem. [Lei] Nikou Lei. If you are interested in her website, go to. url: https: //nikoulei.com.