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