Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
IV.2 The Cohomology R<strong>in</strong>g 163<br />
(for Z is commutative)<br />
=(−1) 1 2 (p+q)(p+q−1) (−1) 1 2 q(q−1) (−1) 1 2 p(p−1)<br />
= c q ({x0,...,xq}) × c p ({xq,...,xp+q})<br />
=(−1) pq c q ({x0,...,xq}) × c p ({xq,...,xp+q}) .�<br />
Hence, H ∗ (K;Z) is a graded skew-commutative r<strong>in</strong>g with identity. We now prove<br />
that H ∗ (−;Z) is a contravariant functor from the category of simplicial complexes<br />
Csim to the category of graded commutative r<strong>in</strong>gs with identity.<br />
(IV.2.4) Lemma. For any c p ∈ C p (L;Z),c q ∈ C q (L;Z), and any simplicial function<br />
f : K → L, the equality<br />
holds.<br />
C p+q ( f )(c p ∪ c q )=C p ( f )(c p ) ∪C q ( f )(c q )<br />
Proof. For each {x0,...,xp,...,xp+q}∈Cp+q(K),<br />
C p+q ( f )(c p ∪ c q )({x0,...,xp,...,xp+q})<br />
=(c p ∪ c q )Cp+q( f )({x0,...,xp,...,xp+q})<br />
�<br />
cp ({ f (x0),..., f (xp)}) × c<br />
=<br />
q ({ f (xp),..., f (xp+q)}) (∀i�= j) f (xi) �= f (x j)<br />
0 otherwise.<br />
On the other hand,<br />
(C p ( f )(c p ) ∪C q ( f )(c q ))({x0,...,xp,...,xp+q})<br />
=(c p Cp( f ) ∪ c q Cq( f ))({x0,...,xp,...,xp+q})<br />
�<br />
cp ({ f (x0),..., f (xp)}) × c<br />
=<br />
q ({ f (xp),..., f (xp+q)}) (∀i�= j) f (xi) �= f (x j)<br />
0 otherwise.<br />
�<br />
Consequently, C ∗ ( f ): C ∗ (L;Z) → C ∗ (K;Z) preserves cup products; s<strong>in</strong>ce the<br />
homomorphisms C p ( f ) commute with the appropriate coboundary operators, the<br />
simplicial function <strong>in</strong>duces a r<strong>in</strong>g homomorphism H ∗ ( f ): H ∗ (L;Z) → H ∗ (K;Z).<br />
Regard<strong>in</strong>g the cohomology of polyhedra, given any polyhedron |K|, wemay<br />
def<strong>in</strong>e <strong>in</strong> H ∗ (|K|;Z) a structure of r<strong>in</strong>g with identity simply because H ∗ (|K|;Z)=<br />
H ∗ (K;Z). We must verify that H ∗ (−;Z) is a contravariant functor from the category<br />
of polyhedra P to the category of graded commutative r<strong>in</strong>gs with identity.<br />
In fact, let f : |K| →|L| be a cont<strong>in</strong>uous function and g: K (r) → L a simplicial<br />
approximation of f . We know that the homomorphism H ∗ (π r ;Z) <strong>in</strong>duced by the<br />
projection π r : K (r) → K is an isomorphism; <strong>in</strong> addition,<br />
H ∗ ( f ;Z)=H ∗ (π r ;Z) −1 H ∗ (g;Z): H ∗ (|L|;Z) → H ∗ (|K|;Z).