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28 Chapter 1 The Fundamental Group
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 basepoint
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 between π1 (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
let h : I→X be a path from x0 to x1 , with the inverse path
h(s) = h(1 − s) from x1 back to x0 . We can then associate h
x
to each loop f based at x 0 x1 f
1 the loop h f h based at x0 .
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 general
n fold product f1 ··· fn in which the path fi is traversed in the time interval
i−1 i
n , n . Either way, we define a change-of-basepoint map βh : π1 (X, x1 )→π1 (X, x0 )
by βh [f ] = [h f h]. This is well-defined since if ft is a homotopy of loops based at
x1 then h ft h is a homotopy of loops based at x0 .
Proposition 1.5. The map β h : π 1 (X, x 1 )→π 1 (X, x 0 ) is an isomorphism.
Proof: We see first that βh is a homomorphism since βh [f g] = [h f g h] =
[h f h h g h] = βh [f ]βh [g]. Further, βh is an isomorphism with inverse βh since
βhβh [f ] = βh [h f h] = [h h f h h] = [f ], and similarly βhβh [f ] = [f ]. ⊔⊓
Thus if X is path-connected, the group π1 (X, x0 ) is, up to isomorphism, independent
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.
Proposition 1.6. A space X is simply-connected iff there is a unique homotopy class
of paths connecting any two points in X .
Proof: 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 the
constant loop and π1 (X, x0 ) = 0. ⊔⊓