The Fundamental Group

IntroductionFundamental Group

ConclusionAddis Ababa University

The Fundamental Group

Miliyon T.

Addis Ababa UniversityDepartment of Mathematics

June 22, 2015

Miliyon T. June 22, 2015 Fundamental Group 1 / 26



”First tell them what you are going to tell them. Tell them.Tell them what you have told them.”

— Paul Halmos, I want to be a mathematician




Henri Poincare

(1854-1912)




Outline

IntroductionBasic Set TheoryBasic Group TheoryBasic TopologyHomeomorphism

Fundamental GroupHomotopyHomotopy of pathsFundamental Group

Conclusion




The basic idea that we need in this section is an equivalencerelation which is defined as follows.

Definition

An equivalence relationa on a set X is a relation R ⊂ X × Xsuch that

Reflexive: (x , x) ∈ R for all x ∈ X .

Symmetric: (x , y) ∈ R implies (y , x) ∈ R.

Transitive: (x , y) and (y , z) ∈ R imply (x , z) ∈ R.

aIt helps us a lot in defining the elements of the fundamental group. As wewill soon enough, it would have been too hard (maybe impossible) for us todefine the elements of the fundamental group without it.




The other important thing we should know is a mathematicalgroup.

Definition (Group)

An algebraic structure with one binary operation (G ,∆) is called agroup iff the following four conditions(group axioms) are satisfied

Closure: ∀a, b ∈ G ⇒ a∆b ∈ G .

Associative: ∀a, b, c ∈ G ⇒ a∆(b∆c) = (a∆b)∆c.

Identity: ∃e ∈ G 3 a∆e = e∆a, ∀a ∈ G .

Inverse: ∀a ∈ G , ∃a−1 ∈ G 3 a∆a−1 = e = a−1∆a.

If ∆ is commutative, then G is called an abelian group. In ourdefinition above there is a word ”algebraic structure”. Which isnothing but a non empty set together with one or more finitaryoperations defined on it. We usually designate a∆b by ab.




The notion of the fundamental group for the first time appears inthe works of Poincare around 1895. He was trying to classifytopological spaces(Riemann Surfaces) in the same time that hediscover this beautiful concept. So, here we are definingTopological space.

Definition

A topology on a set X is a collection τ of subsets of a non emptyset X satisfying the following axioms:

1 ∅ and X are in τ .

2 The union of any number of sets in τ is in τ .

3 The intersection of any two sets in τ is in τ .

The members of τ are then called τ − open sets, or simply opensets.

The ordered pair (X , τ) is called a topological space(TS).




Definition (Continuous Map)

Let X and Y be TS. A map f : X → Y is said to be cont. if foreach open subset V of Y , the set f −1(V ) is an open subset of X .

Lemma

A function f : X → Y is continuous if and only if the inverseimage of every closed subset of Y is a closed subset of X .

Proof.

Suppose f : X → Y is continuous, and A a closed subset of Y .Then A′ is open, and so f −1(A′) is open in X . Butf −1[A′] = (f −1[A])′; therefore f −1[A] is closed. Conversely, assumeA closed in Y implies f −1[A] closed in X . Let G be an open subsetof Y . Then G ′ is closed in Y , and so f −1[G ′] = (f −1[G ])′ is closedin X . Hence, f −1[G ] is open and therefore f is continuous.




Theorem (The pasting lemma)

Let X = A ∪ B, where A and B are closed in X . Let f : A→ Yand g : B → Y be continuous. If f (x) = g(x) for every x ∈ A∪B,then f & g combine to give a continuous function h : X → Ydefined by setting h(x) = f (x) if x ∈ A and h(x) = g(x) if x ∈ B.

Proof.

Let C be a closed subset of Y . Now

h−1(C ) = f −1(C ) ∪ g−1(C ),

by elementary set theory. Since f is continuous, f −1(C ) is closedin A hence closed in X . Similarly, g−1(C ) is closed in B andtherefore closed in X . Their union h−1(C ) is thus closed in X .




Definition (Homeomorphism)

Let X and Y be topological spaces; let f : X → Y be a bijection.If both the function f and the inverse function

f −1 : Y → X

are continuous, then f is called a homeomorphism.




Definition (Path)

Let I = [0, 1], the closed unit interval. A path from a point a to apoint b in a topological space X is a continuous functionf : I → X with f (0) = a and f (1) = b. Here a and b are calledinitial and terminal points respectively.

Figure : Path




Definition (Homotopy)

If f and f ′ are continuous maps of the space X into the space Y ,we say that f is homotopic to f ′ if there is a continuous mapF : X × I → Y such that

F (x , 0) = f (x) and F (x , 1) = f ′(x)

for each x . Where I = [0, 1] the unit interval. The map F is calleda homotopy between f and f ′.

If f is homotopic to f ′ we write f ' f ′. If f ' f ′ and f ′ is aconstant map, we say that f is nulhomotopic. We think of ahomotopy as a continuous one-parameter family of maps from Xto Y . If we imagine the parameter t as representing time, then thehomotopy F represents a continuous deforming of the map f tothe map f ′, as t goes from 0 to 1.




Definition

If f : I → X and f ′ : I → X are two paths with the same initialpoint p ∈ X and the same terminal point q ∈ X . We say that f ishomotopic to f ′ if there is a continuous map F : I × I → X 3

F (s, 0) = f (s) and F (s, 1) = f ′(s),F (0, t) = p and F (1, t) = q

for each s ∈ I and t ∈ I . F is a path homotopy between f and f ′

see figure 2.

Figure : Homotopy of path




Theorem

The relations ' and 'p are equivalence relations.

Proof.(i) Given f it is trivial that f ' f ; the map F (x , t) = f (x) is therequired homotopy. If f is a path, F is a path homotopy.(ii) Given f ' f ′, we show that f ′ ' f . Let F be a homotopybetween f and f ′. Then G (x , t) = F (x , 1− t) is a homotopybetween f ′ and f . If f a path homotopy so is G .(iii) Suppose that f ' f ′ and f ′ ' f ′′. WTS f ' f ′′. Let F be ahomotopy between f and f ′′ and let F ′ be a homotopy between f ′

and f ′′. Define G : X × I → Y by the equation

G (x , t) =

{F (x , 2t), for t ∈ [0, 1/2]

F ′(x , 2t − 1), for t ∈ [1/2, 1]




proof continued...

The map G is well-defined, since if t = 1/2, we have

F (x , 2t) = f ′(x) = F ′(x , 2t − 1)

Because G is continuous on the two closed subsets X × [0, 1/2]and X × [1/2, 1] of X × I , it is continuous on all of X × I , bypasting lemma (6). Thus G is the required homotopy between fand f ”. The following figure illustrates if F and F ′ are pathhomotopies, so is G

Figure : Transitive property of path homotopy. �




Definition (Concatenation)

If f is a path in X from x0 to x1, and if g is a path from x1 to x2,we define the product of f ∗ g of f and g to be the path h given bythe equations

h(s) =

{f (2s), for s ∈ [0, 1/2]

g(2s − 1), for s ∈ [1/2, 1]




Definition

A loop in a topological space is a path in the space whose initialpoint and terminal point are the same. If the initial point andterminal point of a loop in the topological space X are both thepoint x0 ∈ X , we will say that the loop is based at x0.

Let X be a topological space and x0 be a point in X . Then the x0neighborhood of curves in X , C (X , x0), is the collection of allcontinuous mappings f : I → X of the unit interval into X suchthat f (0) = x0 = f (1). i.e. the collection of all loops based at x0.

Definition

Let f and g be two maps in C (X , x0) that means f and g areloops based at x0. Then f is homotopic to g modulo x0 if f and gare homotopic in a usual sense with some additional restriction.Here is the restriction: If H is the homotopy between f and g ,then H(0, t) = x0 = H(1, t).




Theorem

The set of path homotopy equivalence class of loops based atx0 ∈ X is a group under the ”multiplication” defined by[α][β] = [αβ]. This group is denoted by π1(X , x0) and is called thefundamental group of X at x0.

Proof outline.

I. The operation is well defined.

II. Associative

III. The identity element is ex0 .

IV. The inverse of α(t) is α(1− t).




Instead of the fundamental group of X based at x0 it would be niceto have the fundamental group of X . In other words, we would liketo have the fundamental group depend only on the space, and noton the particular point of the space that we base our loops at.

Theorem

Let x0, x1 ∈ X . If there is a path in X from x0 to x1 then thegroups π1(X , x0) and π1(X , x1) are isomorphic.

Proof.To show the two groups are isomorphic is just a matter of finding abijective map from one to the other.Let γ be a path from x0 to x1. If α is a loop based at x0, then(γ−1 ∗ α) ∗ γ is a closed path based at x1.




proof cont...

Figure : The importance of base point

We therefore define uγ : π1(X , x0)→ π1(X , x1)by uγ [α] = [γ−1 ∗ α ∗ γ] that is, follow γ−1 from x1 to x0, thenfollow α around back to xo , then follow γ back to x1, all giving aloop based at x1.




proof cont...

uγ([α] ∗ [β]) = uγ([α ∗ β])

= [γ−1 ∗ α ∗ β ∗ γ]

= [γ−1 ∗ α ∗ γ ∗ γ−1 ∗ β ∗ γ]

= [γ−1 ∗ α ∗ γ] ∗ [γ−1 ∗ β ∗ γ]

= uγ([α]) ∗ uγ([β])

Thus, uγ is a homomorphism.Using the path γ−1 from x1 to x0 we can define

uγ−1 : π1(X , x1)→ π1(X , x0)

by uγ−1([α]) = [γ ∗ α ∗ γ−1].




proof cont...

Now, check

uγ−1uγ [α] = uγ−1 [γ−1 ∗ α ∗ γ] = [γ ∗ γ−1 ∗ α ∗ γ ∗ γ−1] = [α]

uγuγ−1 [α] = uγ [γ ∗ α ∗ γ−1] = [γ−1 ∗ γ ∗ α ∗ γ−1 ∗ γ] = [α]

So, uγ is bijective and hence an isomorphism.�




Conclusion

Determining whether two given topological spaces arehomeomorphic is a fundamental question in topology.

• Showing two space are homeomorphic is a matter ofconstructing a continuous map from one to the other whichhas also a continuous inverse.

• If we can find some topological property that holds for onetopological space but not for the other, then this two spacesare not homeomorphic.

Example

1 [0, 1] is not homeomorphic to (0, 1) since the first is compactand the second is not.

2 R is not homeomorphic to R2 since deleting a point from R2

leaves a connected space and deleting a point from R doesn’t.




Example

i. R2 is not homeomorphic to R3. Because deleting apoint from R3 leaves a simply connected space, butdeleting a point from R2 does not.

ii. S2 � T using similar argument.

As we have seen earlier, the idea of simple connectedness isgeneralised through the fundamental group, which includessimple connectedness as a special case. The condition of simpleconnectedness is just the condition that the fundamental group ofX is the trivial group. So, the most important way of determiningtwo spaces are not homeomorphic is by using their fundamentalgroup.




Just for Fun

A Man is topologically equivalent to torus.

Figure : Drawing by Fuad M.




References

James Munkres

Topology, Second Edition.

I.M singer

Lecture notes in elementary topology and geometry.

Seymour Lipschutz

Genereal Topology (1965).


The Fundamental Group

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