Simplicial Structures in Topology

VI.4 Obstruction Theory 237
Proof. The hypothesis c n+1
f ∼ zn+1 points to the existence of an n-cochain
c n ∈ C n (K,L;πn(W )) such that
c n+1
f − zn+1 = d n (c n ).
We construct g over each n-simplex σ i of K n ∪ L (we may suppose that σ i is not in
L, as indicated by Remark (II.4.8)) as follows. We suppose that c n (σ i )=[hi] rel ∂ I n;
since
we may view hi as a map
(|σ i |,| •
σ i |) ∼ = (I n ,∂I n ),
hi : (|σ i |,| •
σ i |) → (W,w0),
for a suitable w0. On the other hand, the restriction of f n to |K n−1 ∪L| is homotopic
to a constant map, and we note that
In this way, we have constructed a map
[ f n ∗ (hi ∗ f n ) −1 ] rel ∂ I n =[hi] rel ∂ I n = c n (σ i ).
gi = hi ∗ f n : (|σ i |,| •
σ i |) → (W,w0)
whose homotopy class rel∂I n coincides with cn (σ i ). By proceeding like this for
each n-simplex of Kn ∪ L, we obtain an extension g: |Kn+1 ∪ L|→W of the restriction
of f n to |Kn−1 ∪ L| such that cn+1 g = zn+1 . �
We finally prove the most important theorem of this section.
(VI.4.9) Theorem. Let an n-simple space W , a polyhedron |K| with one of its subpolyhedra
|L|, and a map f : |L|→W be given; in addition, let f n−1 : |Kn−1 ∪L|→
W be an extension of f and suppose that f n−1 extends to |Kn ∪ L|. Then, f n−1 can
be extended to |Kn+1 ∪ L| if and only if c n+1
f is cohomologous to zero.
Proof. If f n−1 can be extended to |Kn+1 ∪L|,thenfn−1has an extension g: |Kn ∪ L|
→ W that can be extended to |Kn+1 ∪ L|; therefore, by Theorem (VI.4.3), wehave
cn+1 g = 0. However, c n+1
f ∼ cn+1 g (see Theorem (VI.4.6))andsoc n+1
f ∼ 0.
Reciprocally, if c n+1
f ∼ 0, we conclude that there exists an extension g: |Kn ∪
L| →W of f n−1 such that cn+1 g = 0 (see Theorem (VI.4.8)). Hence, by Theorem
(VI.4.3), it is possible to extend g over the entire polyhedron Kn+1 ∪ L|. �