The fundamental group

Given two spaces X and Y let [X, Y] denote the set of all homotopy classes of maps from X to Y. More generally if (X, A) and (Y, B) are two topological pairs, let [X, A;Y, B] denote the set of all relative homotopy classes of maps from (X, A) to (Y, B). These sets are generally just discrete sets of points. However, under certain hypotheses they are endowed with extra structure. The fundamental group of a space X is a canonical example of this situation. Before we define this construction let us start with some generalities concerning homotopy classes of maps.

The first important feature of [X, Y] is the fact that if f : X $\rightarrow$ X' and g : Y $\rightarrow$ Y' then there are induced maps

f^# : [X', Y] $$\rightarrow$$ [X, Y]      and      g_# : [X, Y] $$\rightarrow$$ [X, Y'],

defined as follows: Let [$\alpha$] $\in$ [X', Y] be the homotopy class of some map $\alpha$ : X' $\rightarrow$ Y. Define f^#[$\alpha$] = [$\alpha$of]. If [$\beta$] $\in$ [X, Y] is the homotopy class of some $\beta$ : X $\rightarrow$ Y, define g_#[$\beta$] = [go$\beta$].

Lemma 6.1   Let f : Z$\rTo$X and g : Y$\rTo$W be maps. Then the induced maps

f^# : $$\left[\vphantom{X,Y}\right.$$X, Y$$\left.\vphantom{X,Y}\right]$$$$\rTo$$$$\left[\vphantom{Z,Y}\right.$$Z, Y$$\left.\vphantom{Z,Y}\right]$$      and      g_# : $$\left[\vphantom{X,Y}\right.$$X, Y$$\left.\vphantom{X,Y}\right]$$$$\rTo$$$$\left[\vphantom{X,W}\right.$$X, W$$\left.\vphantom{X,W}\right]$$

are well defined.

The second and probably most important feature of sets of homotopy classes of maps is their homotopy invariance.

Theorem 6.2   Let f, g : X $\rightarrow$ Y be homotopic maps. Then for any space Z the induced maps

f^# : [Y, Z] $$\rightarrow$$ [X, Z]      and      g^# : [Y, Z] $$\rightarrow$$ [X, Z]

and

f_# : [Z, X] $$\rightarrow$$ [Z, Y]      and      g_# : [Z, X] $$\rightarrow$$ [Z, Y]

coincide.

Corollary 6.3   If f : X $\rightarrow$ X' and g : Y $\rightarrow$ Y' are homotopy equivalences then both f^# : [X', Y] $\rightarrow$ [X, Y] and g_# : [X, Y] $\rightarrow$ [X, Y'] are bijections of sets.

A pointed map from one pointed space to another is simply a map of pairs f : (X, x₀) $\longrightarrow$ (Y, y₀). Also we will sometimes say that X is a pointed space without using the pair notation (X, x₀).

Lemma 6.4   Let (X, x₀),(Y, y₀) and (Z, z₀) be pointed spaces. Then

[X $$\vee$$ Y,(x₀, y₀);Z, z₀] $$\cong$$ [X, x₀;Z, z₀] x [Y, y₀;Z, z₀]

and

[X, x₀;Y x Z,(y₀, z₀)] $$\cong$$ [X, x₀;Y, y₀] x [X, x₀;Z, z₀]

We are now ready to define a sequence of extremely important topological invariants. Namely homotopy groups. However, in this course only one of those groups will be discussed in detail, namely the so called fundamental group.

Definition 6.5   Let (X, x₀) be a pointed space. Consider the n-sphere Sⁿ, n $\geq$ 0 as a pointed space by taking the point (1, 0, ^... , 0) $\in$ Rⁿ as a basepoint b₀. The n-th homotopy group $\pi_{n}^{}$(X, x₀) of (X, x₀) is defined to be the set of relative homotopy classes of maps [Sⁿ, b₀;X, x₀]. In particular $\pi_{1}^{}$(X, x₀) is called the fundamental group of X at x₀.

Exercise 6.6   Show that the set $\pi_{0}^{}$(X, x₀) is the set of path components of X

Let us now think about the fundametal group, in slightly intuitive terms. By definition the fundamental group of a pointed space (X, x₀) is the set of homotopy classes of maps from S¹ to X sending b₀ to x₀. Thus it is the set of homotopy classes of loops in X, which start and end at x₀. How can such a set become a group? If $\lambda$ and $\omega$ are two loops in X starting and ending at x₀ one can imagine ``adding them up'' by defining a new loop, i.e. a map from S¹ to X, which goes half the time along $\lambda$ and the second half along $\omega$. This is an operation which takes two loops and adds them together thus defining a binary operation on the set of all loops in X which start and end at the base point. One could wonder whether this makes this set into a group.

Unfortunately, this is not quite the case. Namely the operation we have just defined is not associative and does not have a unit. Indeed, if you let $\lambda$, $\omega$ and $\nu$ be three loops, multiplying them as ($\lambda$$\omega$)$\nu$ means letting the product loop go the for first quarter along $\lambda$, for the second quarter along $\omega$ and for the last half of the time along $\nu$. On the other hand the product $\lambda$($\omega$$\nu$) goes for the first half through $\lambda$ and then the second half is split in two between $\omega$ and $\nu$. These are simply not the same maps. Also the natural choice for a unit is the constant loop at the base point. But multiplying an arbitrary loop by the constant loop in any order does not give the original loop, but rather a new loop that goes through the original one twice as fast and then stays put for the other half.

It is easy to imagine though why multiplication in this set of loops is associative and has a unit up to homotopy. That is to say, although the two ways of multiplying loops ($\lambda$$\omega$)$\nu$ and $\lambda$($\omega$$\nu$) do not give the same loop do give homotopic loops. Multiplying with the constant loop does not yield the original loop but does give one homotopic to it. Thus if instead of considering the loops themselves we consider homotopy classes of loops, we will get a group. This is the idea behind the definition of the fundamental group, originally due to Poincaré. We shall now proceed by making the discussion more rigorous.

We now explicitly define multiplication in the fundamental group and prove its basic properties.

Consider the circle S¹. There is a map called the ``pinch map''

p : S¹$$\rTo$$S¹ $$\vee$$ S¹

given by identifying the base point with its antipodal. Notice that p is a based map, namely it sends the base point of S¹ to the base point of the wedge. On the other hand for every pointed space X there is a canonical map $\phi$ : X $\vee$ X $\longrightarrow$ X called the folding map, given by sending each factor to X by the identity map.

By Lemma 6.4 for any pointed space (X, x₀) there is an isomorphism of sets

[S¹ $$\vee$$ S¹, b₀;X, x₀] $$\cong$$ $$\pi_{1}^{}$$(X, x₀) x $$\pi_{1}^{}$$(X, x₀).

The operation on $\pi_{1}^{}$(X, x₀) is by definition, the map induced by p. More explicitely, given two homotopy classes [f],[g] $\in$ $\pi_{1}^{}$(X, x₀) represented by pointed loops f, g : S¹ $\longrightarrow$ X, their product [f][g] is the homotopy class of the loop given by the composition

S¹$$\;\stackrel{p}{\longrightarrow}\;$$S¹ $$\vee$$ S¹$$\;\stackrel{f\vee g}{\longrightarrow}\;$$X $$\vee$$ X$$\;\stackrel{\phi}{\longrightarrow}\;$$X.

The next lemma is the major step in proving associativity of this multiplication.

Lemma 6.7   The maps

l : S¹$$\;\stackrel{p}{\longrightarrow}\;$$S¹ $$\vee$$ S¹$$\;\stackrel{p\vee id}{\longrightarrow}\;$$(S¹ $$\vee$$ S¹) $$\vee$$ S¹

and

r : S¹$$\;\stackrel{p}{\longrightarrow}\;$$S¹ $$\vee$$ S¹$$\;\stackrel{id\vee p}{\longrightarrow}\;$$S¹ $$\vee$$ (S¹ $$\vee$$ S¹)

are homotopic.

Now that we have a homotopy between the maps r and l we are ready to prove associativity of multiplication in the fundamental group. Indeed by Lemma 4.18 one has

[S¹ $$\vee$$ S¹ $$\vee$$ S¹, b₀;X, x₀] $$\cong$$ $$\pi_{1}^{}$$(X, x₀) x $$\pi_{1}^{}$$(X, x₀) x $$\pi_{1}^{}$$(X, x₀).

Thus the maps r and l induce two functions

r^*, l^* : $$\pi_{1}^{}$$(X, x₀) x $$\pi_{1}^{}$$(X, x₀) x $$\pi_{1}^{}$$(X, x₀) $$\longrightarrow$$ $$\pi_{1}^{}$$(X, x₀).

If [$\alpha$],[$\beta$],[$\gamma$] $\in$ $\pi_{1}^{}$(X, x₀) then by definition

r^*([$$\alpha$$],[$$\beta$$],[$$\gamma$$]) = [$$\alpha$$]([$$\beta$$][$$\gamma$$])      and      r^*([$$\alpha$$],[$$\beta$$],[$$\gamma$$]) = ([$$\alpha$$][$$\beta$$])([$$\gamma$$]).

But we have shown that r and l are homotopic and therefore the induced maps coincide, proving that multiplication on $\pi_{1}^{}$(X, x₀) is an associative operation.

Before we proceed, let us recapitulate on what was done here. We defined the multiplication on $\pi_{1}^{}$(X, x₀) as the map induced by the pinch map. Then we observed that the two ways of pinching a circle into a wedge of three circles are homotpic. Notice that the proof of this fact depends on the fact that the map f defined in the proof is homotopic to the identity map on I. This map simply reparametrises the partition for the interval which is used to define r into the partition used to define l. Since this gives the two possible ways of multiplying three elements in $\pi_{1}^{}$(X, x₀), we obtained associativity of multiplication. The proofs of existence of a unit and inverses for the multiplication can be done in a very similar fashion. We now prove the existence of a unit following the same lines with a bit less detail.

Let 1 denote the class of the constant map in $\pi_{1}^{}$(X, x₀). We must show that for any pointed loop $\alpha$ in X, 1[$\alpha$] = [$\alpha$]1 = [$\alpha$]. Again consider the reparametrisation of the interval given by

f (t) = $$\left\{\vphantom{\begin{array}{ll} 0 & 0\leq t\leq \frac{1}{2}\\ 2t-1 & \frac{1}{2}\leq t \leq 1 \end{array}}\right.$$$$\begin{array}{ll} 0 & 0\leq t\leq \frac{1}{2}\\ 2t-1 & \frac{1}{2}\leq t \leq 1 \end{array}$$ $$\left.\vphantom{\begin{array}{ll} 0 & 0\leq t\leq \frac{1}{2}\\ 2t-1 & \frac{1}{2}\leq t \leq 1 \end{array}}\right.$$

Given a based loop $\alpha$ : S¹ $\longrightarrow$ X the element 1[$\alpha$] in $\pi_{1}^{}$(X, x₀) is given by the class of the composition $\alpha$(e^2$\pi$if(t)). But f (t) as a self map of the unit interval is homotopic to the identity map relative to the boundary $\partial$I. And so $\alpha$(e^2$\pi$if(t)) is homotopic to $\alpha$(e^2$\pi$it) as a pointed map, showing that 1[$\alpha$] = [$\alpha$]. The proof that 1 is a right unit is similar and is left for the reader.

For the proof of existence of inverses we use a little trick. For any based loop $\alpha$ representing [$\alpha$] in $\pi_{1}^{}$(X, x₀) let [$\alpha$]^-1 be the class represented by the loop $\alpha^{-1}_{}$(e^2$\pi$it) = $\alpha$(e^{2$\pi$i(1 - t)}). Then by definition [$\alpha$][$\alpha$]^-1 is represented by

$$\alpha$$*$$\alpha^{-1}_{}$$(e^2$\pi$it) = $$\left\{\vphantom{\begin{array}{ll} \alpha(e^{2\pi i2t}) & 0\leq ... ... \alpha^{-1}(e^{2\pi i(2t-1)}) & \frac{1}{2}\leq t\leq 1 \end{array}}\right.$$$$\begin{array}{ll} \alpha(e^{2\pi i2t}) & 0\leq t\leq \frac{1}{2}\\ \alpha^{-1}(e^{2\pi i(2t-1)}) & \frac{1}{2}\leq t\leq 1 \end{array}$$ $$\left.\vphantom{\begin{array}{ll} \alpha(e^{2\pi i2t}) & 0\leq t... ... \alpha^{-1}(e^{2\pi i(2t-1)}) & \frac{1}{2}\leq t\leq 1 \end{array}}\right.$$ = $$\left\{\vphantom{\begin{array}{ll} \alpha(e^{2\pi i2t}) & 0\leq ... ...2}\\ \alpha(e^{2\pi i(2-2t)}) & \frac{1}{2}\leq t\leq 1 \end{array}}\right.$$$$\begin{array}{ll} \alpha(e^{2\pi i2t}) & 0\leq t\leq \frac{1}{2}\\ \alpha(e^{2\pi i(2-2t)}) & \frac{1}{2}\leq t\leq 1 \end{array}$$ $$\left.\vphantom{\begin{array}{ll} \alpha(e^{2\pi i2t}) & 0\leq t... ...2}\\ \alpha(e^{2\pi i(2-2t)}) & \frac{1}{2}\leq t\leq 1 \end{array}}\right.$$

Equivalently $\alpha$*$\alpha^{-1}_{}$ can be written the composition

S¹$$\;\stackrel{p}{\longrightarrow}\;$$S¹ $$\vee$$ S¹$$\;\stackrel{1\vee -1}{\longrightarrow}\;$$S¹ $$\vee$$ S¹$$\;\stackrel{\alpha\vee\alpha}{\longrightarrow}\;$$X $$\vee$$ X$$\;\stackrel{\phi}{\longrightarrow}\;$$X.

The reader should have no trouble verifying that the composition $\phi$o($\alpha$ $\vee$ $\alpha$ is identically the same as the composition

S¹ $$\vee$$ S¹$$\;\stackrel{\phi}{\longrightarrow}\;$$S¹$$\;\stackrel{\alpha}{\longrightarrow}\;$$X.

Consequently $\alpha$*$\alpha^{-1}_{}$ can be written as the composition

$$\alpha$$$$\phi$$(1 $$\vee$$ - 1)p

and it suffices to show that $\phi$(1 $\vee$ - 1)p is null-homotopic. But it is easy to see that this composition factors through the unit interval which is a contractible space and hence the map is null-homotopic as required.

To summarise our observation we consider once more what has been done. Notice that multiplication on the fundamental group was defined as the map induced by the pinch map on S¹. Namely p : S¹ $\longrightarrow$ S¹ $\vee$ S¹ induces a map

p^# : [S¹ $$\vee$$ S¹, b₀;X, x₀] $$\longrightarrow$$ [S¹, b₀;X, x₀].

The right hand side is by definition the fundamental group $\pi_{1}^{}$(X, x₀) and by Lemma 4.18 the left hand side is $\pi_{1}^{}$(X, x₀) x $\pi_{1}^{}$(X, x₀). We than used only the features of the pinch map to conclude associativity, the existence of a unit and the existence of inverses. Indeed the fact that the fundamental group of a space X is indeed a group depends entirely on the fact that S¹ admits a pinch map. An important side remark at this point is that every suspension space admits a pinch map, and indeed if X is a suspension space and Y is arbitrary, the existence of the pinch map implied that the set of pointed homotopy classes of maps from X to Y forms a group under multiplication induced by the pinch map.

Our observations have thus produced a group associated to each pointed space (X, x₀). If f : X $\longrightarrow$ Y is a pointed map then one obtains an induced map

f_# : $$\pi_{1}^{}$$(X, x₀) $$\longrightarrow$$ $$\pi_{1}^{}$$(Y, y₀).

The next question one may naturally ask is whether or not this map is a homomorphism.

The approach we took makes it very easy to observe that the answer to this question is positive. Indeed let [$\alpha$],[$\beta$] $\in$ $\pi_{1}^{}$(X, x₀) be represented by maps $\alpha$ and $\beta$ respectively. Then [$\alpha$][$\beta$] is represented by the class of the composition

S¹$$\;\stackrel{p}{\longrightarrow}\;$$S¹ $$\vee$$ S¹$$\;\stackrel{\alpha\vee\beta}{\longrightarrow}\;$$X $$\vee$$ X$$\;\stackrel{\phi}{\longrightarrow}\;$$X.

Then f_#([$\alpha$][$\beta$]) is represented by fo$\phi$o($\alpha$ $\vee$ $\beta$)op. But fo$\phi$ is identically the same as the composite

X $$\vee$$ X$$\;\stackrel{f\vee f}{\longrightarrow}\;$$Y $$\vee$$ Y$$\;\stackrel{\phi}{\longrightarrow}\;$$Y.

Composing the last with ($\alpha$ $\vee$ $\beta$)op one obtains a map representing f_#[$\alpha$]f_#[$\beta$]. And so we conclude that f_# is a homomorphism of groups. Notice that we did not check that the identity element of $\pi_{1}^{}$(X, x₀) is sent to the identity of $\pi_{1}^{}$(Y, y₀) under f_#, but this is obvious. We summarise our results in the following

Theorem 6.8   Let X be a pointed space. Then the set [S¹, b₀;X, x₀] of pointed homotopy classes of maps forms a group under the map induced by the pinch map on S¹. Furthermore, if f : X $\longrightarrow$ Y is a pointed map then the induced map

f_# : $$\pi_{1}^{}$$(X, x₀) $$\longrightarrow$$ $$\pi_{1}^{}$$(Y, y₀)

is a group homomorphism.

In example sheet 5 you are required to show that if X is path-connected then $\pi_{1}^{}$(X, x₀) does not depend of the choice of a base point. It should be remarked though that if x₁ is another point in X, then $\pi_{1}^{}$(X, x₀) is isomorphic to $\pi_{1}^{}$(X, x₁) but the isomorphism involves a choice of a path from x₀ to x₁ and thus is not canonical.

Calculation of the fundamental group of a space is generally not easy. In the example sheet you are required to show that contractible spaces as well as spheres of dimension at least 2 have a trivial fundamental group. Spaces with a trivial fundamental group are called simply-connected or 1-connected. You are also asked to calculate the fundamental group of a product and of a wedge in terms of the fundamental groups of the factors. The first non-trivial example of a non-vanishing fundamental group is given by $\pi_{1}^{}$(S¹). However in order to be able to give a rigorous calculation of it we shall need the machinery of ``covering spaces'' which will be covered in the next section. This will also enable us to compute the fundamental group of many other spaces. We conclude this subsection by an intuitive discussion of the fundamental group of some simple spaces.

Consider the circle S¹ first. Based loops in S¹ are in some sense determined up to homotopy by the number of times the loop completes an entire circle and in what orientation. There are canonical choices for loops which go around the circle exactly n times for any integer n, namely the loops taking z to zⁿ. These loops turn out to represent all the elements in $\pi_{1}^{}$(S¹). Thus $\pi_{1}^{}$(S¹) $\cong$ Z. As we shall see later the fact that S¹ is the orbit space of a R under the natural Z action is closely related to the fundamental group of the circle being the integers.

Exercise 6.9   Calculate the fundamental group of the n-fold torus Tⁿ = S¹ x ^... S¹, n-times and of the wedge S¹ $\vee$ S¹.

Next consider the real projective space RP². This space can be considered as a hemisphere with an identification on its boundary, which gives the prjective plane. The identification indicates that one loop around the boundary is not null-homotopic. However if one continues to loop around once more then the resulting loop is null-homotopic. The outcome is that the fundamental group of RP² is isomorphic to Z/2Z.