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]) = 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 :
X such that
A and
= {x0}.
In particular,
= {x0}, so
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
is constant and so
is essentially
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)
πn(X, x0)
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
× {0}) (X, A) can be extended to a map
(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.
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