0.5. Higher homotopy groups 11
The group operation is defined exactly as in the absolute case. The
only catch is that the coordinate tn plays a special role in the definition of
the relative homotopy group, and the definition of the sum of two equiv-
alence classes uses the coordinate t1 in an independent way. As a re-
sult, the definition of the group operation only makes sense if n ≥ 2;
π1(X, A, x0) is simply considered to be a set of equivalence classes with
no natural group structure (just like π0(X, x0)). As in the absolute case,
a continuous map ψ : (X, A, x0) → (Y, B, y0) induces a homomorphism
ψ∗ : πn(X, A, x0) → πn(Y, B, y0) defined by ψ∗([f]) = [ψ ◦ f].
We may identify the absolute group πn(X, x0) with the relative group
πn(X, {x0},x0), so there is a natural (inclusion-induced) homomorphism
j∗ : πn(X, x0) → πn(X, A, x0). There is also an inclusion-induced homo-
morphism i∗ : πn(A, x0) → πn(X, x0). An element of πn+1(X, A, x0) is
represented by a map f :
In+1
→ X such that
f(In)
⊂ A and
f(Jn)
= {x0}.
In particular,
f(∂In)
= {x0}, so
f|In
may be viewed as representing an el-
ement of πn(A, x0). This correspondence induces a natural homomorphism
∂ : πn+1(X, A, x0) → πn(A, x0). The homomorphism ∂ is called the bound-
ary operator because
f|Jn
is constant and so
f|In
is essentially
f|∂In+1.
The three homomorphisms just described combine to form an extremely
valuable long exact sequence, the homotopy sequence of the pair (X, A) with
base point x0:
· · · − → πn+1(X, A, x0)
∂
− → πn(A, x0)
i∗
− → πn(X, x0)
j∗
− → πn(X, A, x0) − → · · ·
A proof of the exactness of the homotopy sequence of a pair can be found
on pages 344 and 345 of (Hatcher, 2002).
Definition. A topological pair (X, A) is said to be k-connected, k ≥ 0,
if each map
(In
×
{0},∂In
× {0}) → (X, A) can be extended to a map
(In+1,Jn)
→ (X, A) for n = 0, 1,...,k.
The final theorem in the section will be applied in the proof of the basic
engulfing theorem of Chapter 3.
Theorem 0.5.2. Let X be a pathwise connected space with subset A and
basepoint x0 ∈ A. For each k ≥ 0 the following are equivalent.
(1) The pair (X, A) is k-connected.
(2) πn(X, A, x0) = 0 for n ≤ k.
(3) If K is a k-dimensional polyhedron and L is a closed subpolyhedron
of K, then any map (K, L) → (X, A) is homotopic, rel L, to a map
K → A.
Proof. Exercise 0.5.4.
