Simplicial Structures in Topology

IV.2 The Cohomology Ring 161
is such that
(∀c p ∈ C p (K;Z)) c 0 ∪ c p = c p ∪ c 0 = c p .
We conclude from these remarks that the cup product provides the graded Abelian
group C ∗ (K;Z) with a structure of graded ring with identity c 0 .
(IV.2.1) Theorem. For every c p ∈ C p (K;Z) and c q ∈ C q (K;Z),
holds.
∂ p+q (c p ∪ c q )=∂ p (c p ) ∪ c q +(−1) p c p ∪ ∂ q (c q )
Proof. We prove this result by computing ∂ p+q (c p ∪ c q ) on any generator
{x0,...,xp+q+1} of Cp+q+1(K).
∂ p+q (c p ∪ c q )({x0,...,xp+q+1})=(c p ∪ c q )(∂p+q+1({x0,...,xp+q+1}))
=(c p ∪ c q �
p+q+1
) ∑
i=0
On the other hand,
and
Therefore,
∂ p (c p ) ∪ c q ({x0,...,xp+q+1})=
= ∂ p (c p )({x0,...,xp+1}) × c q ({xp+1,...,xp+q+1})
� �
p+1
= c p
∑
= c p ∪ c q
(−1) i {x0,...,�xi,...,xp+1}
i=0
�
p
∑(−1)
i=0
i {x0,...,�xi,...,xp+q+1}
(−1) i �
{x0,...,�xi,...,xp+q+1} ;
× c q ({xp+1,...,xp+q+1})
�
+
+(−1) p+1 c p ({x0,...,xp}) × c q ({xp+1,...,xp+q+1})
c p ∪ ∂ q (c q )({x0,...,xp+q+1})=
= c p ({x0,...,xp}) × c q
�
p+q+1
∑
i=p
= c p ({x0,...,xp}) × c q ({xp+1,...,xp+q+1})+
+(−1) p c p ∪ c q
(−1) i �
{xp,...,�xi,...,xp+q+1}
�
p+q+1
∑ (−1)
i=p+1
i {x0,...,�xi,...,xp+q+1}
�
.
∂ p+q (c p ∪ c q )({x0,...,xp+q+1})=
=(∂ p (c p ) ∪ c q +(−1) p c p ∪ ∂ q (c q ))({x0,...,xp+q+1}) . �