10 0. Prequel

A map ψ : (X, x0) → (Y, y0) leads to an induced homomorphism ψ# :

πn(X, x0) → πn(Y, y0) defined by ψ#([f]) = [ψ ◦ f]. The nth homotopy

group has the expected functoriality properties with respect to such base-

point-preserving maps of spaces. Unlike the fundamental group, however,

πn(X, x0) is always abelian when n 1.

Application of lifting theorems such as (Munkres, 2000, Lemma 79.1)

to a covering map Θ : X → X quickly yields that Θ# : πn(X , x) →

πn(X, Θ(x)) is an isomorphism for n 1. Consequently, πn(X, x0) = 0 for

all n 1 if the universal covering space of X is contractible.

Note that X is path connected if and only if π0(X, x0) = 0 and X is

simply connected if and only if both π0(X, x0) and π1(X, x0) are trivial.

There is a corresponding kind of connectivity in every dimension and it is

detected by the higher homotopy groups.

Definition. A space X is said to be k-connected, k ≥ 0, if each map

∂In+1

→ X can be extended to a map

In+1

→ X for n = 0, 1,...,k.

Theorem 0.5.1. Let X be a pathwise connected space with basepoint x0.

For each k ≥ 0 the following are equivalent.

(1) X is k-connected.

(2) πn(X, 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, x0) is homotopic, rel L, to the

constant map K → x0.

Proof. Exercise 0.5.2.

There is also a local version of connectivity, the usefulness of which will

be illustrated in the next two sections.

Definition. A space X is said to be locally connected in dimension k (ab-

breviated k-LC) if for every x ∈ X and for every neighborhood U of x in X

there exists a neighborhood V of x such that each map

Sk

→ V extends to a

map

Bk+1

→ U. The space X is said to be locally k-connected (abbreviated

LCk)

if X is n-LC for 0 ≤ n ≤ k.

We now define a relative homotopy group πn(X, A, x0), which is associ-

ated with a space X, a nonempty subset A of X, and a basepoint x0 ∈ A. In

order to define the relative group, it is convenient to identify

In−1

with the

face

In−1

×{0} ⊂

∂In

and to use

Jn−1

to denote the union of the remaining

faces of

In

(so

∂In

=

In−1 ∪Jn−1

and

In−1 ∩Jn−1

=

∂In−1).

An element of

πn(X, A, x0) is represented by a map of triples

(In,In−1,Jn−1)

→ (X, A, x0).

Two such maps are equivalent if they are homotopic via a homotopy of

triples.