Simplicial Structures in Topology

234 VI Homotopy Groups
This derives directly from the preceding lemma when we interpret S n+1 as | •
σn+2|
and, writing as usual σn+2 = {x0,...,xn+1}, if we consider the orientation of its
(n + 1)-simplexes given by
(−1) i {x0,...,�xi,...,xn+1}
(see the beginning of Sect. II.2.3). �
The cocycles c n+1
f are of particular interest, as we may realize from what follows.
(VI.4.3) Theorem. An extension f n : |Kn ∪L|→Woff: |L|→W can be extended
to |Kn+1 ∪L| if and only if c n+1
f : Cn+1(K;Z) → πn(W) is the trivial homomorphism.
Proof. The map f n can be extended to |Kn+1 ∪ L| if and only if f n can be extended
to |σn+1|, foreveryσn+1of Kn+1 ∪ L (see Lemma (VI.3.4)); therefore, f n can be
extended to |Kn+1 ∪ L| if and only if c n+1
f = 0. �
Somehow, c n+1
f indicates whether there are obstructions to the extension of f n ;
this is why cn+1 f is known as obstruction cocycle . We now go over some examples
of possible applications of Theorem (VI.4.3).
1. Let a polyhedron |K| with dimension n ≥ 2 be given and let |L| be a subpolyhedron;
if, for every 0 ≤ i ≤ n − 1, πi(Y ) ∼ = 0, then every map f : |L|→Y can
be extended to a map f : |K| →Y. For instance, if Y = S 2 , |K| is the torus
T 2 with the triangulation shown in Sect. III.5.1 and |L| is the geometric realization
of a generating 1-cycle of the homology of T 2 (for example, L is the
simplicial complex with vertices {0}, {3}, {4} and 1-simplexes {0,3}, {0,4},
{3,4}), then every map f : |L| →S 2 can be extended to a map f : T 2 → S 2 .
The construction of f is easy: we choose a point of yi ∈ S 2 for each vertex of
K distinct from {0}, {3}, {4} (for these, we have y0 = f ({0}), y3 = f ({3}),
and y4 = f ({4})); then, since S 2 is path-connected, we choose a path of S 2 for
each 1-simplex of K; in this way, we extend f to f 1 : |K 1 ∪ L|→S 2 ; finally, we
apply Theorem (VI.4.3) to extend f 1 to |K|.
2. (VI.4.4) Theorem. Let K be a simplicial complex of dimension n ≥ 2 and
Y a space such that πi(Y ) ∼ = 0, for every 0 ≤ i ≤ n. Then, any two maps
f ,g: |K|→Y are homotopic.
Proof. Let f ,g: |K|→Y be two maps given arbitrarily. The product |K|×I is
an (n +1)-dimensional polyhedron; let |L| = |K|×∂I and let h: |L|→Y be the
mapsuchthat
h||K|×{0} = f and h| |K|×{1} = g.
For each vertex {x} ∈K we choose a path hx : {x}×I → Y (this is possible
because Y is path-connected) and, in doing so, we obtain an extension h 1 of h
to |K 1 ∪ L|. By Theorem (VI.4.3), we have an extension of h 1 to |K 2 ∪ L|, and
so on, arriving to a homotopy
H : |K|×I → Y
from f to g. �