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]) = 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.
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