Simplicial Structures in Topology

216 VI Homotopy Groups
The group
πn(Y,y0) :=[S n ,Y ]∗
is the nth homotopy group of the based space (Y,y0); the homotopy groups πn(Y,y0)
with n ≥ 2 are also called higher homotopy groups of (Y,y0).
The next lemma characterizes the maps f : (S n ,e0) → (Y,y0) that are homotopic
to the constant map and thus characterizes the unit element of the group πn(Y,y0);
here the reader is asked to return to Exercise 2 of Sect. I.2 to review at least the
definition of contractibility of a space. Note that the sphere S n is homeomorphic
to the geometric realization of the simplicial complex •
σn+1 (σn+1 is an (n + 1)simplex).
(VI.3.4) Lemma. A based map
f : | •
σn+1|−→Y ;
maybeextendedto|σn+1| if and only if f is homotopic to the constant map.
Proof. Let us suppose f to be extended to a map
f : |σn+1|−→Y.
Since |σn+1| ∼ = D n+1 is contractible, the identity map of |σn+1| onto itself is homotopic
to the constant map c; let
H : |σn+1|×I −→ |σn+1|
be the homotopy, which joins 1 |σn+1| and c. The composite map
•
σn+1 × I
is a homotopy from f to c.
Let us now suppose that
ι × 1I ��
|σn+1|×I H ��
|σn+1|
G: | •
σn+1|×I −→ Y
is a homotopy from f to c. Since the pair of polyhedra (|σn+1|,| •
σn+1|) has the
Homotopy Extension Property (see Theorem (III.1.7)), there exists a map
H : |σn+1|×I −→ Y
f
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Y