Simplicial Structures in Topology

110 III Homology of Polyhedra
Let K =(X,Φ) be a simplicial complex. A projection of K (1) on K is a function
π : K (1) → K
that takes each vertex of K (1) (thatistosay,asimplexofK) to one of its vertices.
Any projection is a simplicial function; in fact, if
{σi0 ,···,σin }∈Φ(1) , with σi0 ⊂···⊂σin ,
π({σi0 ,···,σin }) ⊂ σin and, since the latter is a simplex of K, we conclude that
π({σi0 ,···,σin }) ∈ Φ. From the homological point of view, the choice of vertex for
each simplex is absolutely irrelevant because, if π ′ were any other projection, we
would have
π ′ ({σi0 ,···,σin }) ⊂ σin ⊃ π({σi0 ,···,σin })
for every {σi 0 ,··· ,σin }∈Φ(1) ; therefore, the projections π and π ′ would be contiguous.
It follows from these considerations that we may choose π to be the function
that associates to each simplex of K its last vertex.
(III.2.2) Theorem. Let π : K (1) → K be a projection. Then, for every n ∈ Z,
Hn(π,Z) is an isomorphism.
Proof. The projection π produces a chain complex homomorphism
C(π): C(K (1) ) −→ C(K) ;
we wish to find a chain complex homomorphism
ℵ: C(K) −→ C(K (1) )
such that ℵC(π) is chain homotopic to 1 C(K (1) ) and C(π)ℵ is chain homotopic to
1 C(K). If we reach this goal, from the homological point of view, the homomorphism
H∗(ℵ): H∗(K,Z) −→ H∗(K (1) ,Z)
induced by ℵ is the inverse of H∗(π,Z).
The morphism ℵ does not come from a simplicial function and is defined by
induction as follows. Since the vertices of K are also vertices of K (1) ,wedefineℵ0
on the generators {x} of C0(K) by ℵ0({x})={x}. Suppose that we have defined
ℵi for every i = 1,...,n − 1 such that
ℵi−1∂ K
i = ∂ K(1)
ℵi .
We define ℵn on the generators σn of Cn(K) (namely, the oriented n-simplexes of
K) by the formula
ℵn(σn): = ℵn−1(∂ K n (σn)) ∗ b(σn)
i