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