Simplicial Structures in Topology

II.2 Abstract Simplicial Complexes 61
(II.2.16) Definition. An n-chain cn ∈ Cn(K) is an n-cycle (or simply cycle) if
∂n(cn)=0; thus Zn(K) is the set of all n-cycles. An n-chain cn, for which we can
find an (n +1)-chain cn+1 such that cn = ∂n+1(cn+1),isann-boundary; thus, Bn(K)
is the set of all n-boundaries. Two n-chains cn and c ′ n are said to be homologous if
∈ Bn(K).
cn − c ′ n
What we have just described is a method to associate a graded Abelian group
H∗(K;Z)={Hn(K;Z) | n ∈ Z}
to any oriented simplicial complex K ∈ Csim. To define a functor on Csim we must
see what happens to the morphisms; we proceed as follows. Let f : K =(X,Φ) →
L =(Y,Ψ) be a simplicial function (K and L have a fixed orientation). We first
define
Cn( f ): Cn(K) −→ Cn(L)
on the simplexes by
Cn( f )({x0,...,xn})=
� { f (x0),..., f (xn)}, (∀i �= j) f (xi) �= f (x j)
0, otherwise
andthenextendCn( f ) linearly over the whole Abelian group Cn(K). It is easy to
prove that ∂ L n Cn( f )=Cn−1( f )∂ K n ,foreveryn∈Z (one can verify this on a single
n-simplex). We now define
Hn( f ): Hn(K;Z) −→ Hn(L;Z)
z + Bn(K) ↦→ Cn( f )(z)+Bn(L)
for every n ≥ 0. We begin by observing that Cn( f )(z) is a cycle in Cn(L): in fact,
since z is a cycle,
∂ L n Cn( f )(z)=Cn−1( f )∂ K n (z)=0 .
On the other hand, we note that Hn( f ) is well defined: let us assume that z − z ′ =
∂ K n+1 (w); then
Cn( f )(z − z ′ )=Cn( f )∂ K n+1 (w)=∂ L n+1 Cn+1( f )(w)
and thus Cn( f )(z − z ′ ) ∈ Bn(L); from this, we conclude that Hn( f )((z − z ′ )+
Bn(K)) = 0.
If n < 0, we set Hn( f )=0; in this way, we obtain a homomorphism Hn( f )
between Abelian groups, for every n ∈ Z. The reader is invited to prove that
for every n ∈ Z.
Hn(1K)=1 Hn(K) e Hn(gf)=Hn(g)Hn( f )
(II.2.17) Remark. The construction of the homology groups Hn(K;Z) is independent
from the orientation of K, up to isomorphism. In fact, suppose that O and O ′