Simplicial Structures in Topology

IV.1 Cohomology with Coefficients in G 159
and the morphisms
Hom(C(K);Z)={Hom(Cn(K);Z)} ,
Hom(C(K (r) );Z)={Hom(Cn(K (r) );Z)}
Hom(C(π r );Z): Hom(C(K);Z) → Hom(C(K (r) );Z) ,
Hom(ℵ r ;Z): Hom(C(K (r) );Z) → Hom(C(K);Z) .
The chain homotopies mentioned before become homotopies of the functor
Hom(−;Z) and so we obtain homomorphisms
H ∗ (ℵ r ;Z): H ∗ (K (r) ;Z) → H ∗ (K;Z) ,
H ∗ (π r ;Z): H ∗ (K;Z) → H ∗ (K (r) ;Z)
that are isomorphisms and the inverse of each other. We then define
as
H ∗ ( f ;Z): H ∗ (|L|;Z) → H ∗ (|K|;Z)
H ∗ ( f ;Z)=H ∗ (ℵ r ;Z)H ∗ (g;Z) .
Considerations similar to the ones in Corollary (III.2.7) allow us to conclude that
H ∗ ( f ;Z) is well defined.
Finally, we use the short exact sequences
Ext(Hn−1(|K|;Z),G) ��
��
H n (|K|;G)
��
��
Hom(Hn(|K|;Z),G) ,
associated with the polyhedron |K|, for computing the cohomology groups, with
coefficients in Z,ofthetorusT 2 and of the real projective plane RP 2 :
Exercises
H i (T 2 ;Z) ∼ ⎧
Z if i = 0
⎪⎨
Z × Z if i = 1
=
⎪⎩
Z if i = 2
0 ifi�= 0,1,2.
H i (RP 2 ;Z) ∼ ⎧
Z if i = 0
⎪⎨
0 if i = 1
=
Z2 ⎪⎩
if i = 2
0 if i �= 0,1,2.
1. An Abelian group G is said to be divisible if, for every positive integer n and every
g ∈ G, there exists a g ′ ∈ G (not necessarily unique) such that ng ′ = g. The Abelian