Simplicial Structures in Topology

164 IV Cohomology
Since g and π r are simplicial functions, H ∗ (g;Z) and H ∗ (π r ;Z) are ring homomorphisms;
however, since H ∗ (π r ;Z) is an isomorphism, also H ∗ (π r ;Z) −1 preserves
cup products. We conclude that if two polyhedra |K| and |L| are of the same
homotopy type, then their cohomology rings are isomorphic (as rings).
We now wish to prove that the equality of the (co)homology groups is a necessary
but not sufficient condition for the cohomology rings to be isomorphic. To this end,
let us return to the cohomology of the bi-dimensional torus T 2 and of the space
X = S 2 ∨ (S 1 ∨ S 1 ). We have already proved (see Sect. III.4) that these two triangulable
spaces have the same homology and, by the Universal Coefficient Theorem
in Cohomology, also the same cohomology; moreover, we have already seen (see
Sect. III.4 again) that T 2 and X are not homeomorphic. We now compute the cohomology
of T 2 and X once more, by explicitly considering the ring structure.
We begin with T 2 . Let us consider the oriented triangulation of T 2 used to
compute H∗(T 2 ;Z) (see Fig. II.10 in Sect. II.2, p.63). We recall that if z1 1 and
z2 1 are the two 1-cycles
then
z 1 1 = {0,3} + {3,4} + {4,0} and z 2 1 = {0,1} + {1,2} + {2,0},
w 1 1
= {1,6} + {6,5} + {5,8} + {8,7} + {7,2} + {2,1}
is a 1-cycle homologous to z 1 1 (see how H∗(T 2 ,Z) was computed) and
w 2 1 = {3,7} + {7,5} + {5,8} + {8,6} + {6,4} + {4,3}
is a 1-cycle homologous to z 2 1 . For each 1-simplex {i, j} of T 2 ,let
be the function such that
{i, j} ∗ ∈ Hom(C1(T 2 ),Z)
(∀{k,ℓ}∈T 2 ) {i, j} ∗ ({k,ℓ})=
We now perform some calculations:
�
1 if {k,ℓ} = {i, j}
0 if {k,ℓ} �= {i, j}.
1. ∂ 1 {1,6} ∗ ({1,6,5})=({1,6} ∗ ∂2)({1,6,5})={1,6} ∗ (∂2{1,6,5})=1
2. ∂ 1 {1,6} ∗ ({1,2,6})=−1
we conclude that ∂ 1 {1,6} ∗ = {1,6,5} ∗ −{1,2,6} ∗ . Similarly, we find that
∂ 1 {6,5} ∗ = {1,6,5} ∗ −{5,6,8} ∗ , ∂ 1 {5,8} ∗ = {5,8,7} ∗ −{5,6,8} ∗ ,
∂ 1 {8,7} ∗ = {5,8,7} ∗ −{7,8,2} ∗ , ∂ 1 {7,2} ∗ = {7,2,1} ∗ −{7,8,2} ∗ ,
∂ 1 {2,1} ∗ = {7,2,1} ∗ −{1,2,6} ∗ ,