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
→ X can be extended to a map
→ 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
→ V extends to a
→ U. The space X is said to be locally k-connected (abbreviated
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
and to use
to denote the union of the remaining
An element of
πn(X, A, x0) is represented by a map of triples
→ (X, A, x0).
Two such maps are equivalent if they are homotopic via a homotopy of