Simplicial Structures in Topology

236 VI Homotopy Groups
Proof. The first three assertions are direct consequences of the definition of difference
cochain. For proving the 4th one, we take any (n + 1)-simplex
σ = {x0,x1,...,xn+1}
of K and let fσ (respectively, gσ ) be the restriction of f (respectively, g) to| •
σ|; we
intend to prove that
that is to say
{(c n+1
f
− cn+1
g ) − d n (δ n ( f ,g))}(σn+1)=0
[ fσ ] − [gσ] − (Σ n+1
i=0 (−1)i δ n ( f ,g)(σ i )) = 0
where σ i = {x0,x1,...,�xi,...,xn+1}. To obtain this result, we identify S n+1 with the
boundary of I n+2 ∼ = |σ|×I,thatistosay,
S n+1 = ∂ (|σ|×I)=|σ|×{0}∪|σ|×{1}∪(∪ n+1
i=0 |σ i |×I);
after this, we define the map of
given by the union of maps
F : | •
σ|×{0}∪| •
σ|×{1}∪(∪i| •
σ i |×I) → W
f : | •
σ|×{0}→W,
g: | •
σ|×{1}→W,
∪ n+1
i=0 fi ∗ g −1
i : (∪i|σ i |×I) → W
and finally, we apply Lemma (VI.4.1) to F. �
(VI.4.7) Remark. If g is a constant map, we conclude that
[ f || •
σ|]=Σ n+1
i=0 [ f ||σ i |].
This is the so-called Homotopy Addition Theorem; it states that the (based) homotopy
class of a map f : Sn → Y is the sum of the homotopy classes of the restrictions
of f to the geometric n-simplexes of the triangulation of the sphere. It is interesting
to note that the Homotopy Addition Theorem appeared (without proof) in the
literature for the first time in [35]; its first formal proof was written by S-J. Hu [20].
Part 4 of Theorem (VI.4.6) shows that two extensions to |Kn ∪ L| of a
map f : |Kn−1 ∪ L| →W have cohomologous (n + 1)-obstruction cocycles and,
therefore, these cocycles produce the same element of the cohomology group
Hn+1 (K,L;πn(W)) (provided that W be n-simple). We now prove the converse of
this result.
(VI.4.8) Theorem. Let W be an n-simple space and f n : |Kn ∪L|→W an extension
of f whose obstruction (n + 1)-cocycle c n+1
f is cohomologous to a cocycle zn+1 ∈
Cn+1 (K,L;πn(W)). Then, there is an extension g: |Kn+1 ∪L|→W of the restriction
of f n to |Kn−1 ∪ L| such that cn+1 g = zn+1 .