This first section begins with the basic definitions and constructions, and then proceeds quickly to an important calculation, the fundamental group of the circle, using notions developed more fully in§1.3. More systematic methods of calculation are given in§1.2. These are sufficient to show for example that every group is realized as the fundamental group of some space. This idea is exploited in the Additional Topics at the end of the chapter, which give some illustrations of how algebraic facts about groups can be derived topologically, such as the fact that every subgroup of a free group is free.
Before proceeding further we need to verify a technical property:
P
roposition 1.2. The relation of homotopy on paths with fixed endpoints in any space is an equivalence relation.The equivalence class of a path f under the equivalence relation of homotopy will be denoted[f ] and called the homotopy class of f .
P
roof: Reflexivity is evident since f ' f by the constant homotopy ft = f . Symmetry is also easy since if f0 ' f1 via ft, then f1' f0 via the inverse homotopyf1−t. For transitivity, if f0 ' f1 via ft and if f1 = g0 with g0 ' g1via gt, then f0' g1 via the homotopy ht that equals f2t for 0 ≤ t ≤1/2 and g2t−1 for1/2 ≤ t ≤ 1. These two definitions
f0
g1
f1
g0
agree fort=1/2 since we assume f1= g0. Continuity of the associated map H(s, t) = ht(s) comes from the elementary
fact, which will be used frequently without explicit mention, that a function defined on the union of two closed sets is continuous if it is continuous when restricted to each of the closed sets separately. In the case at hand we have H(s, t)= F(s, 2t) for 0 ≤ t ≤1/2 and H(s, t)= G(s, 2t − 1) for1/2 ≤ t ≤ 1 where F and G are the maps I×I
→
X associated to the homotopies ft andgt. SinceH is continuous on I×[0,1/2]and onI×[1/2, 1], it is continuous on I×I . tu
Given two paths f , g : I
→
X such that f (1) = g(0), there is a composition or product path f g that traverses first f and then g , defined by the formulaf g(s)=
(f (2s), 0≤ s ≤1/2 g(2s− 1), 1/2≤ s ≤ 1
Thus f and g are traversed twice as fast in order for f g to be traversed in unit time. This product operation respects homotopy classes
since if f0' f1 and g0 ' g1 via homotopies ft and gt, and iff0(1)= g0(0) so that f0 g0 is defined, thenft gt
f0
g1
f1
g0
is defined and provides a homotopyf0 g0' f1 g1.
In particular, suppose we restrict attention to pathsf : I
→
X with the same start-ing and endstart-ing point f (0) = f (1) = x0 ∈ X . Such paths are called loops, and the common starting and ending point x0 is referred to as thebasepoint. The set of all homotopy classes[f ] of loops f : I→
X at the basepoint x0 is denoted π1(X, x0) .P
roposition 1.3. π1(X, x0) is a group with respect to the product [f ][g]= [f g].This group is called the fundamental group of X at the basepoint x0. We will see in Chapter 4 that π1(X, x0) is the first in a sequence of groups πn(X, x0) , called homotopy groups, which are defined in an entirely analogous fashion using the n dimensional cube In in place of I .
P
roof: By restricting attention to loops with a fixed basepoint x0∈ X we guarantee that the product f g of any two such loops is defined. We have already observed that the homotopy class of f g depends only on the homotopy classes of f and g , so the product[f ][g]= [f g] is well-defined. It remains to verify the three axioms for a group.As a preliminary step, define areparametrization of a path f to be a composi-tion f ϕ where ϕ : I
→
I is any continuous map such that ϕ(0) = 0 and ϕ(1) = 1.Reparametrizing a path preserves its homotopy class sincef ϕ' f via the homotopy f ϕt where ϕt(s) = (1 − t)ϕ(s) + ts so that ϕ0 = ϕ and ϕ1(s) = s . Note that (1− t)ϕ(s) + ts lies between ϕ(s) and s , hence is in I , so the composition f ϕt is defined.
If we are given pathsf , g, h with f (1)= g(0) and g(1) = h(0), then both prod-ucts(f g) h and f (g h) are defined, and f (g h) is a reparametrization
of (f g) h by the piecewise linear function ϕ whose graph is shown in the figure at the right. Hence(f g) h' f (g h). Restricting atten-tion to loops at the basepointx0, this says the product in π1(X, x0) is associative.
Given a pathf : I
→
X , let c be the constant path at f (1) , defined by c(s)= f (1) for alls∈ I . Then f c is a reparametrization of f via the function ϕ whose graph is shown in the first figure at the right, so f c' f . Similarly,c f ' f where c is now the constant path at f (0), using the reparametrization function in the second figure. Taking f to be a loop, we deduce that the homotopy class of the constant path atx0 is a two-sided identity inπ1(X, x0) .
For a path f from x0 tox1, theinverse path f from x1 back to x0 is defined by f (s) = f (1 − s). To see that f f is homotopic to a constant path we use the homotopy ht = ft gt where ft is the path that equals f on the interval [0, 1− t]
and that is stationary atf (1− t) on the interval [1 − t, 1], and gt is the inverse path of ft. We could also describe ht in terms of the associated function
H : I×I
→
X using the decomposition of I×I shown in the figure. On the bottom edge of the squareH is given by f f and below the ‘V’ we let H(s, t) be independent of t , while above the ‘V’ we let H(s, t) beindependent of s . Going back to the first description of ht, we see that since f0= f andf1is the constant pathc at x0,ht is a homotopy fromf f to c c= c . Replacing f by f gives f f ' c for c the constant path at x1. Taking f to be a loop at the basepointx0, we deduce that [ f ] is a two-sided inverse for [f ] in π1(X, x0) . tu
E
xample 1.4. For a convex set X inRnwith basepointx0∈ X we have π1(X, x0)= 0, the trivial group, since any two loops f0 and f1 based at x0 are homotopic via the linear homotopy ft(s)= (1 − t)f0(s)+ tf1(s) , as described in Example 1.1.It is not so easy to show that a space has a nontrivial fundamental group since one must somehow demonstrate the nonexistence of homotopies between certain loops.
We will tackle the simplest example shortly, computing the fundamental group of the circle.
It is natural to ask about the dependence of π1(X, x0) on the choice of the base-point x0. Since π1(X, x0) involves only the path-component of X containing x0, it is clear that we can hope to find a relation betweenπ1(X, x0) and π1(X, x1) for two basepoints x0 and x1 only if x0 and x1 lie in the same path-component of X . So leth : I
→
X be a path from x0 tox1, with the inverse pathh(s)= h(1−s) from x1 back to x0. We can then associate
x1 x0
h to each loop f based at x1 the loop h f h based at x0. f
Strictly speaking, we should choose an order of forming the product h f h , either (h f ) h or h (f h) , but the two choices are homotopic and we are only interested in homotopy classes here. Alternatively, to avoid any ambiguity we could define a gen-eral n fold product f1 ··· fn in which the path fi is traversed in the time interval
i−1
n ,ni .
P
roposition 1.5. The map βh:π1(X, x1)→
π1(X, x0) defined by βh[f ]= [h f h]is an isomorphism.
P
roof: If ft is a homotopy of loops based at x1 then h ft h is a homotopy of loops based at x0, so βh is well-defined. Further, βh is a homomorphism since βh[f g]= [h f g h] = [h f h h g h] = βh[f ]βh[g] . Finally, βhis an isomorphism with inverse βh since βhβh[f ]= βh[h f h]= [h h f h h] = [f ], and similarlyβhβh[f ]= [f ]. tu
Thus if X is path-connected, the group π1(X, x0) is, up to isomorphism, inde-pendent of the choice of basepoint x0. In this case the notation π1(X, x0) is often abbreviated to π1(X) , or one could go further and write just π1X .
In general, a space is called simply-connected if it is path-connected and has trivial fundamental group. The following result explains the name.
P
roposition 1.6. A space X is simply-connected iff there is a unique homotopy class of paths connecting any two points in X .P
roof: Path-connectedness is the existence of paths connecting every pair of points, so we need be concerned only with the uniqueness of connecting paths. Suppose π1(X)= 0. If f and g are two paths from x0 to x1, then f ' f g g ' g since the loops g g and f g are each homotopic to constant loops, using the assumption π1(X, x0) = 0 in the latter case. Conversely, if there is only one homotopy class of paths connecting a basepoint x0 to itself, then all loops at x0 are homotopic to theconstant loop and π1(X, x0)= 0. tu