Simplicial Structures in Topology

VI.4 Obstruction Theory 235
3. (VI.4.5) Theorem. Let K be a simplicial complex of dimension n ≥ 2. Then,
|K| is contractible if and only if πi(|K|) ∼ = 0, for every 0 ≤ i ≤ n.
Proof. If |K| is contractible, the statement is evident. If |K| is n-simple and
πi(|K|) ∼ = 0forevery0≤ i ≤ n, then, by the preceding result, the identity map
1 |K| and any constant map from |K| onto itself are homotopic. �
The condition cn+1 f = 0 is unfortunately too strong for the more general cases,
even if it works well in cases as the ones previously mentioned. The results obtained
when the obstruction cocycles are cohomologous to 0 are much more interesting.
Let us consider these cases. For the next definition (when necessary), we consider
the homotopy group πn(W) with the structure given by Theorem (VI.3.23). Let
f ,g: |Kn ∪ L| →W be two maps whose restrictions to |Kn−1 ∪ L| coincide; in addition,
let fi and gi be the restrictions of f and g to |σ i n|, the geometric realization
of the simplicial complex generated by an n-simplex σ i n of Kn ∪ L. We identify the
space |σ i n | with the hypercube In and interpret fi and gi as maps In → W. Sincefi
and gi coincide at | •
σ i n |≡∂In , the restriction of the map
fi ∗ g −1
i (x1,...,xn)=
�
fi(2x1,...,xn) 0 ≤ x1 ≤ 1 2
gi((2 − 2x1),...,xn) 1
2 ≤ x1 ≤ 1
to ∂I n is homotopic to a constant map, and so
for a suitable w0. Wedefine
fi ∗ g −1
i : (I n ,∂ I n ) −→ (W,w0)
δ n ( f ,g)(σ i n ) :=[fi ∗ g −1
i ] rel∂ I n ∈ πn(W).
We point out to the reader that had σ i n been in L, then
We have thus defined a homomorphism
δ n ( f ,g)(σ i n )=0.
δ n ( f ,g): Cn(K;Z) −→ πn(W )
δ n ( f ,g){Σimiσ i n } := Σimi[ fi ∗ g −1
i ] rel ∂ In; δ n ( f ,g) ∈ Cn (K;πn(W)) is the difference n-cochain of f and g.
(VI.4.6) Theorem. If the maps f ,g,h: |K n ∪ L|→W coincide in |K n−1 ∪ L|, then
1. δ n (g, f )=−δ n ( f ,g)
2. δ n ( f , f )=0
3. δ n ( f ,g)+δ n (g,h)=δ n ( f ,h)
4. d n (δ n ( f ,g)) = c n+1
f
− cn+1 g