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.