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34 Chapter 1 The Fundamental Group
The relation between the fundamental group of a product space and the fundamental
groups of its factors is as simple as one could wish:
Proposition 1.12. π1 (X ×Y) is isomorphic to π1 (X)×π 1 (Y ) if X and Y are pathconnected.
Proof: A basic property of the product topology is that a map f : Z→X ×Y is continuous
iff the maps g : Z→X and h : Z→Y defined by f(z) = (g(z), h(z)) are both
continuous. Hence a loop f in X ×Y based at (x0 ,y0 ) is equivalent to a pair of loops
g in X and h in Y based at x0 and y0 respectively. Similarly, a homotopy ft of a loop
in X ×Y is equivalent to a pair of homotopies gt and ht of the corresponding loops
in X and Y . Thus we obtain a bijection π1 X ×Y,(x0 ,y0 ) ≈ π1 (X, x0 )×π1 (Y , y0 ),
[f ] ↦ ([g], [h]). This is obviously a group homomorphism, and hence an isomorphism.
⊔⊓
Example 1.13: The Torus. By the proposition we have an isomorphism π1 (S 1 ×S 1 ) ≈
Z×Z. Under this isomorphism a pair (p, q) ∈ Z×Z corresponds to a loop that winds
p times around one S 1 factor of the torus and q times around the
other S 1 factor, for example the loop ωpq (s) = (ωp (s), ωq (s)).
Interestingly, this loop can be knotted, as the figure shows for
the case p = 3, q = 2. The knots that arise in this fashion, the
so-called torus knots, are studied in Example 1.24.
More generally, the n dimensional torus, which is the product of n circles, has
fundamental group isomorphic to the product of n copies of Z. This follows by
induction on n.
Induced Homomorphisms
Suppose ϕ : X→Y is a map taking the basepoint x0 ∈ X to the basepoint y0 ∈ Y .
For brevity we write ϕ : (X, x0 )→(Y , y0 ) in this situation. Then ϕ induces a homo-
morphism ϕ∗ : π1 (X, x0 )→π1 (Y , y0 ), defined by composing loops f : I→X based at
x0 with ϕ , that is, ϕ∗ [f ] = [ϕf ]. This induced map ϕ∗ is well-defined since a
homotopy ft of loops based at x0 yields a composed homotopy ϕft of loops based
at y0 ,soϕ∗ [f0 ] = [ϕf0 ] = [ϕf1 ] = ϕ∗ [f1 ]. Furthermore, ϕ∗ is a homomorphism
since ϕ(f g) = (ϕf ) (ϕg), both functions having the value ϕf (2s) for 0 ≤ s ≤ 1 / 2
and the value ϕg(2s − 1) for 1 / 2 ≤ s ≤ 1.
Two basic properties of induced homomorphisms are:
(ϕψ) ∗ = ϕ∗ψ∗ for a composition (X, x0 ) ψ
→ (Y , y0 ) ϕ
→ (Z, z0 ).
1∗ = 1, which is a concise way of saying that the identity map 1:X→Xinduces the identity map 1:π1 (X, x0 )→π1 (X, x0 ).
The first of these follows from the fact that composition of maps is associative, so
(ϕψ)f = ϕ(ψf ), and the second is obvious. These two properties of induced homomorphisms
are what makes the fundamental group a functor. The formal definition