Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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IV.1 Cohomology with Coefficients <strong>in</strong> G 155<br />
for which the follow<strong>in</strong>g sequence of cohomology groups<br />
...→ H n (K;Z) Hn (i)<br />
−→ H n (L;Z) ˜ λ n<br />
−→ H n+1 (K,L;Z) q∗ (n+1)<br />
−→ H n+1 (K;Z) → ...<br />
is exact.<br />
The proof of this theorem follows the steps taken <strong>in</strong> prov<strong>in</strong>g the correspond<strong>in</strong>g<br />
theorem <strong>in</strong> homology Theorem (II.3.1); we only wish to po<strong>in</strong>t out that the ma<strong>in</strong><br />
result needed for this proof is the follow<strong>in</strong>g lemma:<br />
(IV.1.4) Lemma. If<br />
A ��<br />
f<br />
��<br />
B<br />
is a short exact sequence of free groups and G is an Abelian group, then also the<br />
sequence<br />
Hom(C,G) ��<br />
�g<br />
g<br />
��<br />
Hom(B,G)<br />
��<br />
��<br />
C<br />
�f<br />
��<br />
��<br />
Hom(A,G)<br />
is exact.<br />
The reader may take the proof of Lemma (II.5.1) as a basis for prov<strong>in</strong>g this one;<br />
actually, the only tensor product property used <strong>in</strong> Lemma (II.5.1) is that −⊗G is<br />
a functor which takes sums of morphisms <strong>in</strong>to sums of morphisms; this very same<br />
property holds also for the (contravariant) functor Hom(−,G).<br />
The cohomology determ<strong>in</strong>ed by an oriented cha<strong>in</strong> simplex K with coefficients <strong>in</strong><br />
an Abelian group G is def<strong>in</strong>ed <strong>in</strong> the same manner as the one with coefficients <strong>in</strong><br />
Z, except that we apply to the cha<strong>in</strong> complex (C(K),∂) the contravariant functor<br />
Hom(−,G) <strong>in</strong>stead of Hom(−,Z).<br />
We wish to determ<strong>in</strong>e the cohomology with coefficients <strong>in</strong> G of a simplicial complex,<br />
based on its homology with coefficients <strong>in</strong> Z and the Abelian group G. We<br />
seek to use the same previous ideas to our benefit, bear<strong>in</strong>g <strong>in</strong> m<strong>in</strong>d that the functor<br />
A ↦→ A ⊗ G is covariant, whereas A ↦→ Hom(A,G) is contravariant. We beg<strong>in</strong> by review<strong>in</strong>g<br />
what has been done so far <strong>in</strong> general terms. A cocha<strong>in</strong> complex (or cocomplex)<br />
is a graded Abelian group {Cn } together with an endomorphism of degree +1,<br />
called coboundary homomorphism ∂ ∗ = {∂ n : Cn → Cn+1 ,},forwhich∂n+1∂n = 0.<br />
A cocha<strong>in</strong> homomorphism between two cocha<strong>in</strong> complexes (C,∂ ∗ ) and (C ′ ,∂ ′∗ ) is<br />
a homomorphism between the graded Abelian groups C and C ′ that commutes with<br />
the coboundary homomorphisms.<br />
Also the other concepts def<strong>in</strong>ed for cha<strong>in</strong> complexes have their dual correspondents:<br />
the elements of Cn are called n-cocha<strong>in</strong>s; the elements of<br />
are n-cocycles, and those of<br />
Z n (C) := ker(∂ n : C n → C n+1 )<br />
B n (C) := im(∂ n−1 : C n−1 → C n )<br />
are n-coboundaries; f<strong>in</strong>ally, the quotient H n (C) := Z n (C)/B n (C) is the cohomology<br />
group of the cocomplex (C,∂ ∗ ) <strong>in</strong> dimension n.