Simplicial Structures in Topology

62 II Simplicial Complexes
are two distinct orientations of K and denote by KO and KO′ the complex K together
with the orientations O and O ′ , respectively.
The simplexes of KO are denoted by σ, and those of KO′ ,byσ ′ . Now define
φn : Cn(K O ) → Cn(K O′ ) as the function taking a simplex σn into the simplex σ ′ n
if O and O ′ give the same orientation to σn, and taking σn into −σ ′ n if O and O′
give opposite orientations to σn; next, extend φn by linearity over the whole group
Cn(K O ). It is easy to prove that φn is a group isomorphism. Moreover, for every
n ∈ Z, ∂nφn = φn−1∂n. For a given n-simplex σn of KO , we have two cases to
consider:
Case 1: O and O ′ give the same orientation to σn; then
∂nφn(σn)=∂n(σ ′ n)=
n
∑
(−1) i σ ′ n−1,i
i=0
n
φn−1∂n(σn)=φn−1( ∑(−1)
i=0
i n
σn−1,i)= ∑
i=0
Case 2: O and O ′ give different orientations to σn; then
∂nφn(σn)=∂n(−σ ′ n)=
n
∑
(−1) i+1 σ ′ n−1,i
i=0
n
φn−1∂n(σn)=φn−1( ∑(−1)
i=0
i+1 n
σn−1,i)= ∑
i=0
(−1) i σ ′ n−1,i ;
(−1) i+1 σ ′ n−1,i.
Similar to what we did to define the homomorphism Hn( f ), we can prove that φn
induces a homomorphism
which is actually an isomorphism.
Hn(φn): Hn(K O ;Z) −→ Hn(K O′
;Z)
Therefore, up to isomorphism, the orientation given to a simplicial complex has
no influence on the definition of the group Hn(K;Z); thus, we forget the orientation
(however, we note that in certain questions it cannot be ignored). With this, we
define the covariant functor
by setting
H∗(−;Z): Csim −→ Ab Z
H∗(K;Z)={Hn(K;Z) | n ∈ Z} and H∗( f )={Hn( f ) | n ∈ Z}
on objects and morphisms, respectively. The graduate Abelian group H∗(K;Z) is
the (simplicial) homology of K with coefficients in Z.
We are going to compute the homology groups of the simplicial complex T 2
depicted in Fig. II.10 and whose geometric realization is the two-dimensional torus.
We begin by orienting T 2 so that we go clockwise around the boundary of each
2-simplex. To simplify the notation, let us write Ci(T 2 ) as Ci (the same for the