§ Snake lemma

§ Why homomorphisms for chain maps?

First of all, to define a mapping between simplicial complexes {Gi}\{ G_i \} and {Hi}\{ H_i \}, one might naively assume that we can ask for functions {fi:GiHi}\{ f_i: G_i \rightarrow H_i \}:
       ∂    ∂
    G3 → G2 → G1 → 0
    |    |    |
    f    g    h
    ↓    ↓    ↓  
0 → H3 → H2 → H1
      ∂    ∂
Unfortunately, to be able to use the machinery of Homology, we need the {fi}\{ f_i \} to be abelian group homomorphisms. However, this is no great loss. Intuitively, when we want to map complexes, we first say where the generators of the abelian group (Z\mathbb Z-module) maps to; Everything else is determined by the generators. This aligns nicely with our intuition of what a map between complexes should look like: we tell where the geometry goes ("this edge goes there"), and the algebra is "dragged along for the ride". This gives us the diagram:
    G3--∂-→G2--∂-→G1 
    |      |      |
    f3     f2     f1
    ↓      ↓      ↓
0 →H3--∂-→H2--∂-→H1
where the fi are homomorphisms. So, this means we can talk about kernels and images!
    Ker(f3)----→Ker(f2)--→Ker(f1)
       |         |          |
       ↓         ↓          ↓
       G3--∂----→G2----∂---→G1--→ 0 
       |         |          |
       f3        f2         f1
       ↓         ↓          ↓  
    Im(f3)--∂--→Im(f2)--∂-→Im(f1)
F → E → V → {0}
{0} → {e} → {v} → {0} 
The Snake Lemma gives us a mapping d:Ker(f1)Im(f3)d: Ker(f1) \rightarrow Im(f3) such that this long exact sequence is saatisfied:

§ What do we wish to compute?

  • Now that we've agreed that this family of maps {fi:GiHi}\{ f_i : G_i \rightarrow H_i \}ought to be structured maps, the next question is "OK, now what? What doesone want to determine"? Ideally, we would get a new chain complex whichI tacitly denote as {f(Gi)}\{ f(G_i) \}, consisting of the image of GiG_i insideHiH_i and the ability to determine its structure.
  • However, this is the boring bit. We don't really care about the chain complex {f(Gi)}\{ f(G_i) \} per se.What we actually care about are the homology groups! So we would really like a toolthat allows us to compute Hi(f(G))H_i(f(G)) in some convenient fashion.