Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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VI.4 Obstruction Theory 233<br />
c n+1<br />
f : Cn+1(K;Z) −→ πn(W)<br />
that takes each ∑i miσ i ∈ Cn+1(K;Z) <strong>in</strong>to ∑i mi[ f n<br />
σ i]. We notice that if σ ∈ Ψ, then<br />
by Lemma (VI.3.4), [ f n σ ]=0: <strong>in</strong> fact, f is def<strong>in</strong>ed <strong>in</strong> |L| and f n can be therefore<br />
extended to |σn+1|. This allows us to conclude that c n+1<br />
f ∈ Cn+1 (K,L;πn(W)), that<br />
is to say, c n+1<br />
f is a cocha<strong>in</strong>.<br />
The next lemma is useful for prov<strong>in</strong>g that c n+1<br />
f is a cocycle.<br />
(VI.4.1) Lemma. Let W be an n-simple space and S n+1 be the sphere viewed as an<br />
(n + 1)-manifold, which is triangulated by a simplicial complex K =(X,Φ), as<strong>in</strong><br />
Theorem (V.1.5);letσ i , with i = 1,2,...,s, be the (n + 1)-simplexes of K; f<strong>in</strong>ally, let<br />
K n =(X,Φ n ) be the simplicial n-dimensional subcomplex of K, where<br />
Then, for any map f : |K n |→W,<br />
Φ n = {σ ∈ Φ | dimσ ≤ n}.<br />
s<br />
∑<br />
i=1<br />
[ f i σ ]=0.<br />
Proof. By Theorem (V.1.5), everyn-simplex of K is a face of exactly two (n + 1)simplexes;<br />
besides, by Def<strong>in</strong>ition (V.3.1), each n-simplex <strong>in</strong>herits opposite orientations<br />
from its two adjacent (n + 1)-simplexes. The spaces | •<br />
σ i | are homeomorphic<br />
to the sphere Sn whose elements may be considered as t ∧x, with x ∈ Sn−1 ;thisway<br />
of view<strong>in</strong>g the elements of Sn gives us an idea of the orientation of the sphere; <strong>in</strong><br />
other words, if we take the elements <strong>in</strong> the format t ∧ x, we travel the sphere with a<br />
“positive” orientation but, if we take them <strong>in</strong> the format (1−t)∧x,wetravelSnwith the opposite orientation, that is to say, we give Sn a “negative” orientation. With this<br />
<strong>in</strong> m<strong>in</strong>d, we observe that the function f is applied twice on each | •<br />
σ i |: once, viewed<br />
as the function f (t ∧ x) and once, as the function f ((1 − t) ∧ x); on the other hand,<br />
the product by νn<br />
× of the homotopy classes of these functions is the trivial class of<br />
πn(W ) (see Theorem (VI.3.3)).<br />
We note that the base po<strong>in</strong>t of each homotopy class is irrelevant because W is<br />
n-simple. �<br />
(VI.4.2) Theorem. The cocha<strong>in</strong> c n+1<br />
f<br />
is a cocycle.<br />
Proof. It is necessary to prove that, for every (n + 2)-simplex σn+2 of K n ∪ L,<br />
d n+1 (c n+1<br />
f )(σn+2)=c n+1<br />
f (dn+2(σn+2)) = 0.