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.
II.3 Homological Algebra 65<br />
II.3 Introduction to Homological Algebra<br />
In the previous section, we have seen that we can associate a graded Abelian<br />
group C(K) ={Cn(K)} with any simplicial complex K and a homomorphism<br />
∂n : Cn(K) → Cn−1(K), such that ∂n−1∂n = 0, to each <strong>in</strong>teger n; these homomorphisms<br />
def<strong>in</strong>e a graded Abelian group H∗(K;Z) ={Hn(K;Z)}. All this can be<br />
viewed <strong>in</strong> the framework of a more general and more useful context.<br />
A cha<strong>in</strong> complex (C,∂) is a graded Abelian group C = {Cn} together with<br />
an endomorphism ∂ = {∂n} of degree −1, called boundary homomorphism3 ∂ =<br />
{∂n : Cn → Cn−1}, such that ∂ 2 = 0; this means that, for every n ∈ Z, ∂n∂n+1 = 0.<br />
Hence<br />
Bn = im∂n+1 ⊂ Zn = ker∂n<br />
and so we can def<strong>in</strong>e the graded Abelian group<br />
H∗(C)={Hn(C)=Zn/Bn | n ∈ Z};<br />
this is the homology of C.<br />
A cha<strong>in</strong> homomorphism between two cha<strong>in</strong> complexes (C,∂) and (C ′ ,∂ ′ ) is a<br />
graded group homomorphism f = { fn : Cn → C ′ n} of degree 0 commut<strong>in</strong>g with the<br />
boundary homomorphism, that is to say, for every n ∈ Z, fn−1∂n = ∂ ′ n fn.<br />
Cha<strong>in</strong> complexes and cha<strong>in</strong> homomorphisms form a category C, thecategory of<br />
cha<strong>in</strong> complexes.<br />
It is costumary to visualize cha<strong>in</strong> complexes as diagrams<br />
···<br />
��<br />
Cn+1<br />
∂n+1 ��<br />
Cn<br />
and their morphisms as commutative diagrams<br />
···<br />
···<br />
��<br />
Cn+1<br />
��<br />
��<br />
′<br />
C n+1<br />
fn+1<br />
∂n+1 ��<br />
Cn<br />
��<br />
∂ ′ n+1 ��<br />
′<br />
C n<br />
fn<br />
∂n ��<br />
Cn−1<br />
∂n ��<br />
Cn−1<br />
��<br />
∂ ′ n ��<br />
′<br />
C n−1<br />
fn−1<br />
��<br />
···<br />
��<br />
···<br />
��<br />
···<br />
The previous def<strong>in</strong>itions are clearly <strong>in</strong>spired by what we did to def<strong>in</strong>e the homology<br />
groups of a simplicial complex; <strong>in</strong>deed, we emphasize the fact that, for every<br />
simplicial complex X, the graded Abelian group {Cn(K)|n ∈ Z} together with its<br />
boundary homomorphism ∂ K = {∂ K n |n ∈ Z} is a cha<strong>in</strong> complex (C(K),∂ K ). The<br />
cha<strong>in</strong> complex C(K) is said to be positive because its terms of negative <strong>in</strong>dex are 0.<br />
In particular, for every simplicial function f : K → M, the homomorphism<br />
is a cha<strong>in</strong> homomorphism.<br />
C( f ): C(K) → C(M)<br />
3 In some textbooks, it is called differential operator.