The fundamental group

Scribe: Pavithran S Iyer∗

Minutes of Discussion - 1, IISER - Mohali

Presented & Discussed on: 19th Aug, 2011

1 Topological spaces

Let us take a set X with a topology τ defined∗ on it. The set, along with the topology is calleda topological space and is represented by (X, τ). For the sake of convenience, we would in futuredenote the topological space (X, τ) by X. A topological space X is said to be disconnected if itcan be expressed as a disjoint union of subsets Vα where Vα ⊂ X. Else, the topological space X iscalled a connected space.

A function between two topological spaces between X and Y is a set theoretic (many-to-one)map whose domain are elements of X and image is are elements of Y .

If X and Y are two topological spaces, function f : X → Y is said to be continuous iff forany open set† V ⊂ Y , its inverse image: f−1(V ) = {f−1(y)|y ∈ V } is open‡. In otherwords, nomatter how small the neighborhood (open set containing f(x)) V of f(x) is, ∃ a neighborhood Uof x such that f(U) ⊂ V .

The elements of X are called points. A path in X is a continuous function f : I → X whereI = [0, 1]. The points f(0) and f(1) are termed as the end points of the path. Two points x0 and

∗Helpful insights from Prof. Arvind, Prof. Sanjeev, Debabrata, Debmalaya, Amol and Shruti; IISER - Mohali∗This means that τ ⊂ 2X , with the following properties:

1. Empty set is contained in τ : φ ∈ τ

2. τ is closed under union: if {Vα} ∈ τ , then⋃α

Vα ∈ τ .

3. Closed under finite intersection: Vα ∩ Vβ ∈ τ ∀Vα, Vβ ∈ τ

For quick examples, we have the following Topological Spaces:

• X = {1, 2, 3, 4} and τ = {{}, {2}, {1, 2}, {2, 3}, {1, 2, 3, 4}}

• X = R and τ = set of all open intervals in R

†Notice that in this case, continuity cannot be defined by specifying ε and δ neighborhoods around the element.This is because we haven’t defined any notion of “distance” between two elements in sets X and Y . However, thenotion of an open set is often used in such cases to provide the notion of “nearness” between two elements withouthaving to explicitly define any metric on the topological space.‡Formally, one says that f : X → Y is continuous at x ∈ X iff for any neighborhood V ⊂ Y of f(x), ∃ a

neighborhood U ⊂ X of x such that f(U) ⊂ V .

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(a) A path between two points x0 and x1in a (path-connected) topological spaceX is represented as a continuous linejoining x0 and x1. In future, we willbe referring topological spaces which arepath connected.

(b) Another ex-ample of a pathconnected space.Notice that pathconnected spacesneed not havethe propertythat any twopaths can bemoved into oneanother contin-uously (withoutbreaking).

(c) If the space is not connected, thena path between two end-points can havedisconnected pieces too.

Figure 1: Pictorial representation of topological spaces and paths

x1 in X ae said to be path connected iff ∃ a continuous function f : I → X such that f(0) = x0and f(1) = x1. If any two points in X are path connected, then the space X itself is said to bepath connected.

We now give some pictorially representations of topological spaces and paths between pointsin that space:

2 Homotopy and equivalence

We will now be studying different paths on a some topological space X (which we will assumeto be path-connected). It is evident that there can be too many paths between two endpoints tostudy each path individually. Instead, we categorize the paths into various classes, stating thatpaths belonging to a particular class are “essentially the same” and study these classes. Firstly,we need some notion of saying how two paths are essentially the same. Informally or pictoriallystating, two paths α(t) and β(t) which can be continuously deformed into one another (withoutbreaking the path) are considered essentially the same. It is important to note that the two pointsmust share the same end-points. In otherwords, one can give a continuously family of paths (eachpath characterized by a continuous parameter s ∈ [0, 1]) {αs(t)}s∈[0,1] between the same end-points(for t = 0 and t = 1, for any s the function must give the same endpoints), beginning with theα(t) path (at s = 0) and ending with β(t) path (at s = 1). Two such paths α(t) and β(t) arecalled homotopic, denoted by α ∼ β. This is the precise notion of “essentially the same”.

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Figure 2: The square represents the I × I interval with each horizontal line (corresponding eachvalue of s) represents (or is mapped to) a particular path in X between α(0) and α(1). Theconditions H(t, 0) = α(t) and H(t, 1) = β(t) ensure that the bottom and the top sides are mappedto the α and β paths (respectively). Similarly, each of the intermediate horizontal lines for valuesof s between 0 and 1 correspond of a continuous family of paths between the endpoints. All thepoints on the t = 0 and t = 1 lines are mapped to α(0) and α(1) respectively.

Definition We say that two paths α : I → X and β : I → X are homotopic if ∃ a continuousfunction H : I × I → X such that H(0, s) = α(0) = β(0) ∀s ∈ [0, 1], H(1, s) = α(1) = β(1)∀s ∈ [0, 1], H(t, 0) = α(t) and H(t, 1) = β(t). Such a function H is called a homotopy.

As remarked, the homotopy H describes a family of paths between α and β. To show thatα ∼ β, we need to show the existence of a homotopy which is usually done by defining it. Beforeseeing how one defines a homotopy given two paths sharing the same endpoints (which can bedeformed into one another), let us look at a pictorial representation of a homotopy H for twopaths α ∼ β.

Hence to show that α ∼ β, it suffices to diagrammatically represent the concerned homotopyH(t, s) as shown in the figure above.

We now justify how we can say that two paths which are homotopic are “essentially the same”and we do this by showing that homopoty is indeed an equivalence relation. As per the definition§

of an equivalence relation. In each of the cases, we will draw the diagrammatic representation ofthe homotopy.

1. α ∼ α: we need to define a continuous function H(t, s) such that H(t, 0) = α and H(t, 1) = α.The most naive way, which suffices here is: H(t, s) = α(t) ∀s ∈ [0, 1] . In the corresponding

§Any relation ∗ between elements of a set X which satisfies the following properties (∀a, b, c ∈ X) is called anequivalence relation. 1. a ∗ a 2. If a ∗ b, then b ∗ a 3. If a ∗ b and b ∗ c, then a ∗ c

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diagram for H, each of the intermediate lines in the square will be mapped to the same pathα. Needless to say, H(t, s) is continuous in s as well as t.

Figure 3: A constant homotopy.

2. α ∼ β ⇒ β ∼ α: Given a homotopy H between α and β, we need to describe a homotopy Kbetween β and α. Pictorially, the homotopy diagram for H need to be “flipped vertically”(thereby exchanging the bottom and top sides representing α and β respectively.), whichis nothing but redefining the value at s to that at 1 − s, to obtain a diagram for K. Wetherefore define K(t, s) = H(t, 1− s) which is automatically continuous as H is continuous.

Figure 4: H is the given Homotopy and K is the required homotopy. Notice that K(t, 0) = β(t)and K(1, t) = α(t) and hence K is a homotopy for β ∼ α.

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3. α ∼ β & β ∼ γ ⇒ α ∼ γ: Suppose the homotopies for α ∼ β and β ∼ γ are H and Krespectively. We need to construct a new homotopy L for α ∼ γ with the property thatit is continuous, L(t, 0) = α(t) and L(t, 1) = γ(t). To obtain a diagram for L, we need tojust combine the diagrams for H and K by “gluing” the edges representing the β paths withthe diagram for K above that for F . Additionally, before we glue the diagrams, we needto appropriately scale the axis since L must be defined only on I × I. Hence we have adiagram for L. Furthermore, one can even obtain an expression for L in terms of H and K:

L(t, s) =

H(t, 2s) when 0 ≤ s ≤ 1

2

K(t, 2s− 1) when1

2≤ s ≤ 1

.

Figure 5: H and K are the given Homotopies and L is the required homotopy. Notice thatL(t, 0) = α(t) and L(1, t) = γ(t), β is excluded in the boundary conditions. Hence L is a homotopyfor α ∼ γ.

Hence we have established that the homotopy is indeed an equivalence relation.Having defined the notion of paths and their equivalence, we now define a product operation

between two paths in X. If α and β are two paths in X such that “ β begins where α ends”, thatis: β(0) = α(1), we can define a product operation giving γ(t) = α(t)× β(t) to be another path.

Definition A product of two paths α and β, such that β(0) = α(1), is defined to be another pathγ, such that:

γ(t) = α(t)× β(t) =

α(2t) when 0 ≤ t ≤ 1

2

β(2t− 1)& when1

2≤ t ≤ 1

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Figure 6: The product α× β is a path which is obtained by concatenating α and β.

Pictorially, we obtain a product path by simply concatenating any two paths.We now have a small result which relates the product of two pairs of paths with the product

of corresponding homotopic paths:

Lemma 2.1 If α ∼ α′ and β ∼ β′, then α× β ∼ α′ × β′.

Proof Let H and K be homotopies corresponding to α ∼ α′ and β ∼ β′ respectively. We needto derive a homotopy L corresponding to α × β ∼ α′ × β′. As before, we will attempt togive a diagram for L, using those for H and K. Clearly L must satisfy the boundary conditions:L(t, 0) = α× β and L(t, 1) = α′× β′. In otherwords, using the definition for the product of paths,

L(t, 0) =

α(2t) when 0 ≤ t ≤ 1

2

β(2t− 1) when1

2≤ t1

and similarly L(t, 1) =

α′(2t) when 0 ≤ t ≤ 1

2

β′(2t− 1) when1

2≤ t1

.

Just as in the proof of transitivity property of the homotopy equivalence (step: 3 on the precedingpage), we now need to combine the diagrams for H and K by “glue” the t = 1 edge of H witht = 0 edge of K, and appropriately rescaling the axis. One can also give an explicit expression

for the resulting homotopy: L(t, s) =

H(2t, s) when 0 ≤ t ≤ 1

2

K(2t− 1, s) when1

2≤ t1

. Needless to say, L is a

continuous function of its parameters.

Figure 7: H and K are the given Homotopies and L is the required homotopy. The product of twopaths is just their concatenation and hence the representation in the diagram too is a concatenationof the respective intervals with a suitable rescaling. Notice that both L(t, 0) as well as L(t, 1) arethe results of concatenation of two paths, which are α, β in the first case and α′, β′ in the second.

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Hence we have a homotopy corresponding to α× β ∼ α′ × β′ .

Having defined an equivalence relation, the homotopy, we now define an equivalence class orthe homotopy class of (a path) α(t) to be the set of all paths which are homotopic to α. Thehomotopy class of α is denoted by [α].

Definition The homotopy class of α, denoted by [α] = β|β ∼ α. In otherwords, [β] ≈ ¶[α] iffα ∼ β .

One can immediately see that it is necessary for all paths in the equivalence class of α to have thesame end-points as α. Analogous to the multiplication operation defined for two paths α and β(such that α(0) = β(1)), one can define the operation for the respective homotopy classes [α] and[β].

Definition The product of two homotopy classes [α] and [β], such that α(1) = β(0), is defined tobe‖ the homotopy class [α× β].

Proposition 2.2 If α ∼ α′ and β ∼ β′, then [α′]× [β′] = [α]× [β] ≈ [α× β] .

¶We will later see why an equality cannot be an appropriate relation.‖Notice that when we write [α] × [β] = [α × β], the class [α] × [β] includes all loops which begin at α(0) and

end at β(1), but also pass through α(1). On the contrary, [α × β] is the set of all paths which too begin and endat α(0) and β(1) (respectively), but there is however no additional constraint that they must pass through α(1).Hence there are clearly several paths in [α× β] which cannot be included into [α]× [β].

Figure 8: Notice that the product of equivalence classes contains paths which have to pass throughα(1) whereas the equivalence class of the product is a larger class containing paths which do notnecessarily pass through α(1). In this figure, paths σ and ρ are two among ones which are in notincluded in [α]× [β] but they are part of [α× β].

One can therefore look at this equality in two ways:

1. We can take this to be the definition of multiplication of two equivalence classes.

2. Instead of the equality relation between [α× β] and [α]× [β], (which implies that they must have the samecardinality) we use a weaker condition that these two are related unto homotopy, denoting the relation by′ ≈′. In otherwords, when mean that all paths in [α× β] are homotopic to all paths in [α]× [β]. This is thepoint of view we have adopted here.

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3 The fundamental group

Let us consider the special case of a path where α(0) = α(1). Such a path is called a loop and thepoint α(0) is called its basepoint. A further special case of a loop is called a constant loop, definedby α(t) = α0 ∀t ∈ [0, 1]. This loop, denoted by α0, is essentially a point in X, but technically, it isconsidered a path as well since it is trivially a continuous function. Any loop which is homotopicto α0 is called null homotopic.

Figure 9: Various loops based at α(0) which are in different homotopy classes. The multiplicationoperation can now be “freely” defined on these homotopy classes, without giving any attention tothe common endpoint condition. This is because every loop starts and end at the same point andhence trivially one starts where the other ends.

If we take all loops based at some point x0 ∈ X, then they fall in different equivalence or homo-topy classes and one can define a multiplication operation (as before) between any two homotopyclasses.

Theorem 3.1 The set of all homotopy classes of loops based at x0 in a topological space X, formsa group under the multiplication operation ′×′ defined on the equivalence classes. This group,denoted by π(X, x0), is called the Fundamental group of X based at x0.

Proof To show that π(X, x0) forms a group under the product operation for homotopy classes,we have to show the five properties which any set and an operation (defined on its elements) mustsatisfy.

1. Closure: This property is satisfied since the product of any two equivalence classes of yieldsanother equivalence class, by definition.

2. Existence of Identity: Let us represent the equivalence class of the constant loop (basedat x0) as [α0]. From the definition of the product of two equivalence classes, we see that[α0]× [α] ≈ [α0 × α]. We can now define a homotopy from α to α × α0 thereby stating thecorresponding equivalence classes are indeed the same and hence showing the presence of theidentity element in π(X, x0). As before, we will provide the diagram for such a homotopyH, satisfying the boundary conditions: H(t, 0) = α(t) and H(t, 1) = α0 × α(t) .

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Figure 10: The homotopy diagram must be interpreted as corresponding to a family of curveswhere, as s increases, more and more time is spent at the point α0 and finally at s = 1, half thetime is spent at x0 and the other half along the α loop (but twice as fast as at s = 0), givingα0 × α. One can also write an expression for this homotopy, but in this case, it is unnecessarysince we just need to show its existence.

We therefore have an identity element in π(X, x0), which we denote as [α0].

3. Existence of Inverse: Having obtained the identity element as [α0], given [α], it suffices toshow the existence of some element [β] such that: [α] × [β] = [α0]. In otherwords, we needto show that α× β ∼ α0 . Pictorially, α0 is just a point and any loop α can be concatenatedwith any homotopic loop in the “reverse direction” (say α−1(t) = α(1− t)), thereby makingthe resulting loop nullhomotopic. It suffices to define a homotopy H corresponding to:α× α−1 ∼ α0, for which as before, we will give a diagram.

Figure 11: The homotopy diagram must interpreted as corresponding to a family of loops where,as s increases, lesser and lesser time is spent in going along α(t) and back along α−1(t), finallyending up spending all the at x0. One can again write an expression for this homotopy, which isnot currently necessary.

We therefore have an inverse for every element in π(X, x0).

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4. Associativity: We need to show that ([α]× [β])× [γ] = [α]× ([β]× [γ]). For this, it sufficesto show that (α× β)× γ ∼ α× (β × γ), which can be shown by providing a (diagram for a)homotopy H such that: H(t, 0) = (α× β)× γ and H(t, 1) = α× (β × γ).

Figure 12: One can again write an expression for this homotopy, which is not currently necessary.

We have therefore shown that the product operation defined on the elements of π(X, x0) isassociative.

By showing that ψ(X, x0), along with the product operation between equivalence classes ′×′ sat-isfies the group axioms, we therefore conclude that π(X, x0) is a group.

The fundamental group of a topological space (which is path connected) also seems on dependon the basepoint of the loops. We will now show that for a path connected topological space, thebasepoint of the loop is not important in determining the fundamental group. By this, we meanthat the fundamental groups for any two basepoints which are connected by a path, are identicalor isomorphic.

Lemma 3.2 If X is a path connected topological space and x0, x1 ∈ X, then π(X, x0) ' π(X, x1).

Proof To show that π(X, x0) ' π(X, x1), it suffices to show that any mapping which takesequivalence classes of loops based at x0 to a class of loops based at x1 is an isomorphism. Let α0

be the constant loop based at x0, h(t) be a path connecting x1 and x0 and h−1(t) be the path (inthe reverse direction) connecting x0 and x1, such that the product: h−1(t) × h(t) ∼ α0 . Noticethat if α is some loop based at x0, then (h(t)×α(t))× h−1(t) would be a loop based at x1. It nowsuffices to show that the map βh : π(X, x0)→ π(X, x1) defined by: βh[h] = [(h(t)×α(t))×h−1(t)]is an isomorphism (which is a homomorphism with a unique inverse).

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Figure 13: The same path h(t) can be labeled as its inverse, but here a different (but homotopic)path is considered for clarity.

If βh is a homomorphism, then we must show that for any two equivalence classes [f ] and [g]of loops based at x0: βh[f × g] = βh[f ]× βh[g]. Using the definition of the βh map, we have:

LHS = βh[f × g] = [(h× (f × g))× h−1] (by definition)

= [(h× f)× (g × h−1)] = [h× f × g × h−1] (associativity), ignoring brackets for clarity.

= [h× f × h−1 × h× g × h−1] (since h−1h ∼ α0, which corresponds to the identity element)

= [h× f × h−1]× [h× g × h−1] (multiplication operation for equivalence classes)

= βh[f ]× βh[g] (by definition of the βh map)

We are now left with showing that βh has a unique inverse, which we claim is βh−1 . If it holdsthat: βh−1βh[f ] = βhβh−1 [f ] = [f ], then βh−1 can be considered as the inverse map and be denotedas: βh−1 = β−1h .

βh−1βh[f ] = βh−1 [h× f × h−1] (by definition of βh map)

= [h−1 × h× f × h−1 × h] (by definition of βh−1 analogous to βh)

= [f ] (since h−1h = hh−1 = α0)

Similarly: βhβh−1 [f ] = βh[h−1 × f × h] (by definition of βh−1)

= [h× h−1 × f × h× h−1] by definition of βh map

= [f ] (since h−1h = hh−1 = α0)

Hence we have shown that βh is indeed an isomorphism and as a result, π(X, x0) ' π(X, x1) .

As a consequence of the above result, when we talk about the fundamental group of any path con-nected space X, we need not mention any basepoint as a parameter. We express the fundamentalgroup of X as: π(X).

If the fundamental group of a space is trivial, i,e; has only one element: π(X, x0) = [α0], whereα0 is the constant loop based at x0, then it means that all loops based at x0 belong to the sameequivalence class. In otherwords, any loop based at x0 is null-homotopic and moreover, the entirespace can be “shrunk” continuously to the basepoint x0 (or into any other point on X for thatmatter).

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Definition A topological space which is path connected and has the trivial fundamental group,is called simply-connected.

Lemma 3.3 If X is a simply connected space, then there exists a unique homotopy class of pathsconnecting any two points in X.

Proof Let f and g be any two paths between endpoints x0, x1 and α0 be a constant loop based atx0. It suffices to show that f ∼ g. Since X is simply connected, we have π(X, x0) = 0 and henceall loops based at x0 belong to the same equivalence class: f × f−1 ∼ f × g−1 ∼ g× f−1 ∼ g× g−1. Using these relations, we get:

f ∼ f × (g−1 × g) (since g−1 × g = α0 and α× α0 ∼ α.)

∼ (f × g−1)× g (associativity)

∼ (f × f−1)× g (all loops based at x0 belong to the same class: f × f−1 ∼ f × g−1)∼ g (since f × f−1 = α0)

Hence we have shown that an arbitrary pair of paths connecting two points are homotopic, therebyimplying that there is only one homotopy class of paths between any two points in X.

References

[1] [BOOK]: Topology, by Munkres, J.R., Prentice Hall, 2000, http://books.google.co.in/

books?id=XjoZAQAAIAAJ

[2] [BOOK]: (Chapter 2) Algebraic Topology, by Allen Hatcher, http://www.math.cornell.edu/

~hatcher/AT/ATpage.html

[3] [Video Lectures]: Lectures on Algebraic Topology, by Norman Walber, UNSW, http://www.youtube.com/watch?v=J7--sI4A6D0, http://www.youtube.com/watch?v=FZNqUIjPO24

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