§ Snake lemma

§ Why homomorphisms for chain maps?

First of all, to define a mapping between simplicial complexes $\{ G_i \}$ and $\{ H_i \}$, one might naively assume that we can ask for functions $\{ f_i: G_i \rightarrow H_i \}$:

       ∂    ∂
    G3 → G2 → G1 → 0
    |    |    |
    f    g    h
    ↓    ↓    ↓
0 → H3 → H2 → H1
      ∂    ∂

Unfortunately, to be able to use the machinery of Homology, we need the $\{ f_i \}$ 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 ( $\mathbb Z$-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 $d: Ker(f1) \rightarrow Im(f3)$ such that this long exact sequence is saatisfied:

§ What do we wish to compute?

• Now that we've agreed that this family of maps $\{ f_i : G_i \rightarrow H_i \}$ought to be structured maps, the next question is "OK, now what? What does one want to determine"? Ideally, we would get a new chain complex which I tacitly denote as $\{ f(G_i) \}$, consisting of the image of $G_i$ inside $H_i$ and the ability to determine its structure.

• However, this is the boring bit. We don't really care about the chain complex $\{ f(G_i) \}$ per se. What we actually care about are the homology groups! So we would really like a tool that allows us to compute $H_i(f(G))$ in some convenient fashion.